Recent zbMATH articles in MSC 35https://zbmath.org/atom/cc/352021-11-25T18:46:10.358925ZWerkzeugOn the constant scalar curvature Kähler metrics. I: A priori estimateshttps://zbmath.org/1472.140422021-11-25T18:46:10.358925Z"Chen, Xiuxiong"https://zbmath.org/authors/?q=ai:chen.xiuxiong"Cheng, Jingrui"https://zbmath.org/authors/?q=ai:cheng.jingruiThis groundbreaking paper is a technical tour de force on the constant scalar curvature Kähler (CSCK) equation. The basic problem is Calabi's dream of finding canonical metric representatives inside positive line bundle classes. A prototype result is Yau's solution to the Calabi conjecture, and the Chen-Donaldson-Sun solution of the Kähler-Einstein equation in the Fano case subject to K-stability. While a formal infinite dimensional GIT framework has long pointed towards CSCK metrics as the natural generality to consider the canonical metric problem, there are significant technical hurdles to go beyond the Kähler-Einstein case, most notably because the CSCK equation is 4th order instead of second order, and because one loses a priori Ricci curvature bounds, which are essential for applying Cheeger-Colding theory. The goal of this paper is to address the central PDE difficulties. Its techniques are relatively classical, involving maximum principles, Moser style iterations, and Alexandrov maximum principles; the presentation is largely self contained, and accessible to those with basic knowledge of Kähler geometry. However, the application of these techniques are often very clever, and exploit a number of rather delicate cancellation effects which can only be appreciated through substantial calculations.
The 4th order CSCK equation on a compact Kähler manifold can be written as a second order coupled system. The authors consider a slightly more general system (needed for their further work on the continuity method)
\[
\log \det(g_{i\bar{j}}+ \phi_{i\bar{j}}) = F + \log \det(g_{i\bar{j}}),\ \Delta_\phi F = -f + Tr_\phi \eta. \tag{1}
\]
When $f = -R$ (the average Ricci scalar) and $\eta = Ric_g$ this is the CSCK equation. The first equation is complex Monge-Ampère, and the second amounts to a prescription of scalar curvature. The main result of this paper is that under an a priori entropy bound $\int e^FFd\mathrm{vol}_g \leq C$, then the solution to this PDE system is bounded to all derivatives. The entropy bound is natural to the problem, because the CSCK equation is the critical point of the Mabuchi functional (also called the K-energy), which can be written as the entropy term plus a well behaved pluripotential term. The stability condition should be thought of as a coercivity condition on the Mabuchi functional, which will essentially force a bound on the entropy as explained in the subsequent works of the authors in the series.
Some highlights of this paper are:
\begin{itemize}
\item In Section 5, the authors prove a $C^0$-estimate on the Kähler potential from the entropy bound. The main ingredient is an ingenious application of the Alexandrov maximum principle, with the additional input of the Skoda inequality. This is inspired by earlier work of Blocki. Even though the method is quite classical, this result is surprisingly strong, especially in the light of Kolodziej's celebrated $L^\infty$-potential estimate from an a priori $L^p$ bound on $F$ with $p > 1$. This part is of significant independent interest in Kähler geometry, especially the analysis of complex Monge-Ampère.
\item In Section 2, they prove among others things an a priori gradient bound $|\nabla \phi |^2e^{-F} \leq C$. This uses a maximum principle argument, which involves a delicate cancellation effect to knock out some bad mixed derivative terms in the Laplacian.
\item In Section 3, they prove a $W^{2,p}$ type estimate $\int e^{-\alpha(p)F}(n+ \Delta \phi )^pd\mathrm{vol}_g \leq C(p)$ for any exponent $p > 0$. This involves integration by part and an iteration argument. The key is that one can gain exponents on $n+ \Delta \phi$ from the nonlinearity, and any derivative terms of $F$ from the Laplacian computation can be either estimated away in complete squares, or absorbed into the equation for $\Delta F$ which is then a priori controlled.
\item In Section 4, they prove simultaneously $|\nabla F| \leq C$ and $n + \Delta \phi \leq C$, whence the metric has $C^2$ bounds, which reduces the CSCK problem to well known higher order estimates. This is proved by a Moser iteration style argument, based on the $W^{2,p}$ estimate established earlier. The reason for $|\nabla F|^2$ to feature in the proof of the metric upper bound $n + \Delta \phi \leq C$, is that one needs the Laplacian of $|\nabla F|^2$ to provide good Hessian terms. The maximum principle quantity involves another subtle cancellation effect to knock out some bad mixed derivative terms. The reason for the Moser iteration to work, is that Sobolev inequality improves the Lebesgue exponent by a definite magnifying factor > 1, while the fact that \(p\) can be arbitrarily large in the $W^{2,p}$ estimate, ultimately ensures that the application of Hölder inequality can only worsen the Lebesgue exponent by a factor which is arbitrarily close to one, so in the end the improvement effect will win over.
\end{itemize}Blow-up solutions of Liouville's equation and quasi-normalityhttps://zbmath.org/1472.300152021-11-25T18:46:10.358925Z"Grahl, Jürgen"https://zbmath.org/authors/?q=ai:grahl.jurgen"Kraus, Daniela"https://zbmath.org/authors/?q=ai:kraus.daniela"Roth, Oliver"https://zbmath.org/authors/?q=ai:roth.oliverLet \(D\) be a domain in the complex plane and \(C > 0\). Let \(\mathcal{F}_C \) be the set of all functions \(f\) meromorphic in \(D\) for which the spherical area of \( f(D)\) on the Riemann sphere is at most \(C \pi\). Then it is shown that \(\mathcal{F}_C \) is quasi-normal of order at most \( C\). In particular, for every sequence \(\{ f_m \} \) in \(\mathcal{F}_C \) (after taking a subsequence), there is an \( f \) in \(\mathcal{F}_C\) such that (1) or (2) below holds.
(1) \(\{ f_m \}\) converges locally uniformly in \(D\) to \( f \);
(2) There exists a finite nonempty set \(S \subset D\) with at most \(C\) points for which (2a) \(\{ f_m \}\) converges locally uniformly in \(D \backslash S\) to \(f\), and for each \(p\) in \(S\) there exists a sequence \(\{ z_m \}\) in \(D\) such that \(\{z_m \}\) converges to \(p\) and \(\{ f_m^{\#} (z_m) \}\) converges to \(+\infty\); and (2b) for each \(p\) in \(S\) there exists a real number \(\alpha_p \geq 1\) such that in the measure theoretic sense
\[\frac{1}{\pi}(f_m^{\#})^2\text{ converges to } \sum_{p\in S}\alpha_p\delta_p+\frac{1}{\pi}(f^{\#})^2.\]
The authors note that the above may be viewed as extending to all meromorphic functions in \(\mathcal{F}_C\) some well-known work of \textit{H. Brézis} and \textit{F. Merle} [Commun. Partial Differ. Equations 16, No. 8--9, 1223--1253 (1991; Zbl 0746.35006)] on solutions of \(-\Delta u =4e^{2u}\) for locally univalent meromorphic functions. In the comparison (2a) may be seen to correspond with ``Bubbling'', while (2b) corresponds with ``Mass Concentration'' in the Brezis-Merle work. Section 2 of the current manuscript contains a lengthy set of remarks and questions (including open questions) regarding the above comparison, while Section 3 on quasi-normality observes a criterion of Montel and Valiron may be applied to obtain \(\mathcal{F}_C\) quasi-normal. Also introduced in Section 3 is an extension of the Montel-Valiron criterion for quasi-normality where exceptional values are replaced by exceptional functions allowed to depend on the individual members of the family.On linearly independent solutions of the homogeneous Schwarz problemhttps://zbmath.org/1472.300202021-11-25T18:46:10.358925Z"Nikolaev, V. G."https://zbmath.org/authors/?q=ai:nikolaev.vladimir-gennadevichSummary: We study the homogeneous Schwarz problem for Douglis analytic functions. We consider two-dimensional matrices \(J\) with a multiple eigenvalue and a eigenvector, which is not proportional to a real vector. We obtain a sufficient condition for the matrix \(J\) under which there exist two linearly independent solutions of the problem defined in a certain domain \(D\). We present an example.Higher order Dirichlet-type problems in 2D complex quaternionic analysishttps://zbmath.org/1472.300242021-11-25T18:46:10.358925Z"Schneider, B."https://zbmath.org/authors/?q=ai:schneider.berthold.1|schneider.bruce|schneier.bruce|schneider.barry-i|schneider.brandon|schneider.bernd|schneider.baruch|schneider.brit|schneider.bernhard|schneider.barbaraThe author studies a higher-order Dirichlet boundary value problem for solutions of the two-dimensional Helmoltz equation in \(\mathbb R^2\).
In Section 1, Introduction, the author poses the problem to be solved.
In Section 2, Preliminaries, the author recalls the following:
\begin{enumerate}
\item[(1)] The Dirac operator in \(\mathbb R^2\): \[\displaystyle \mathcal D = \frac{\partial }{\partial x } e_1 + \frac{\partial }{\partial y } e_2 ;\]
\item[(2)] The Helmholtz operator in \(\mathbb R^2\): \(\displaystyle \Delta_{\mathbb R^2} + \lambda \mathcal I \), where \(\mathcal I\) denotes the identity operator and \(\lambda \in \mathbb C \setminus \{ 0 \}\);
\item[(3)] The factorization \[ \displaystyle - \left( \mathcal D + \alpha \mathcal I \right) \left( \mathcal D - \alpha \mathcal I \right) = \Delta_{\mathbb R^2} + \lambda \mathcal I , \] where \(\alpha \) is a complex number such that \(\alpha^2 = \lambda\);
\item[(4)] The perturbed Dirac operator: \(\mathcal D_\alpha := \mathcal D + \alpha \mathcal I \).
\item[(5)] The set of \(\alpha\)-hyperholomorphic functions on \(\Omega \subset \mathbb R^2\), i.e., the set of functions defined on \(\Omega\) such that \(\mathcal D_\alpha f =0\).
\item[(6)] The fundamental solution \(\Theta_\alpha^{(M)} (z)\), of the operator \( \displaystyle \left( \Delta_{\mathbb R^2} + \alpha^2 \right)^M\) in \(\mathbb R^2\), where \(M \in \mathbb N\) and the Cauchy Kernel \(\displaystyle \mathcal K_\alpha (z) := - \mathcal D_{- \alpha} \left[ \Theta_\alpha^{(1)} (z) \right] \);
\end{enumerate}
In Section 3.1, the author reminds us the definitions and main properties of the Teodorescu integral operator, the Cauchy integral operator and the singular integral operator.
In Section 3.2 the author examines an orthogonal decomposition of the Sobolev space \(\displaystyle \mathbf W^k_2 \left( \Omega , \, \mathbb H ( \mathbb C) \right) \), \(k \in \mathbb N \cup \{ 0 \}\) with respect to the high-order Dirac operator \(\mathcal D_\alpha^k\). Using the formula
\[ \displaystyle \mathcal D \Theta_\alpha^{ (M)} (z) = \frac{1}{ 2 (M-1) } \Theta_\alpha^{ (M - 1)} (z) , \quad M \in \mathbb N, \; \; M \not= 1, \; \; \forall z \in \mathbb R^2 \setminus \{ 0 \} , \]
a formula for \(\mathcal D_\alpha \Theta_\alpha^{(M)}\) is obtained. Hence two orthogonal decompositions of the Sobolev space \(\displaystyle \mathbf W^k_2 \left( \Omega , \, \mathbb H ( \mathbb C) \right) \) are given in Theorem 3.8.
In Theorem 3.11, it is proved that given \(f \in \mathbf W_2^k (\Omega, \mathbb H ( \mathbb C))\), \(g_0 \in \mathbf W_2^{k + \frac{7}{2}} (\Gamma, \mathbb H ( \mathbb C))\), and \(g_1 \in \mathbf W_2^{k + \frac{3}{2} } (\Gamma, \mathbb H ( \mathbb C))\), \(k \geq 0\), then the following boundary value problem has a unique solution \(h \in \mathbf W_2^{k +4} (\Omega, \mathbb H ( \mathbb C))\):
\[
\left\{ \begin{array}{l} \left( - \Delta_{\mathbb R^2} + 2 \alpha \mathcal D + \alpha^2 \right)^2 h = f \quad \text{in} \quad \Omega , \\
h= g_0 \quad \text{on} \quad \Gamma , \\
\left( - \Delta_{\mathbb R^2} + 2 \alpha \mathcal D + \alpha^2 \right) h = g_1 \quad \text{on} \quad \Gamma. \end{array} \right.
\]
At the end of the paper, in Theorem 3.13, it is proved that, given \(f \in \mathbf W_2^k (\Omega, \mathbb H ( \mathbb C))\) and \(g_p \in \mathbf W_2^{k + 2M \frac{4p +1 }{2}} (\Gamma, \mathbb H ( \mathbb C))\), then the following boundary value problem of higher order has a unique solution \(h \in \mathbf W_2^{k +2M} (\Omega, \mathbb H ( \mathbb C))\):
\[
\left\{ \begin{array}{l} \left( - \Delta_{\mathbb R^2} + \lambda \right)^M h = f \quad \text{in} \quad \Omega , \\
h= g_0 \quad \text{on} \quad \Gamma , \\
\left( - \Delta_{\mathbb R^2} + \lambda \right) h = g_1 \quad \text{on} \quad \Gamma , \\
\vdots \\
\left( - \Delta_{\mathbb R^2} + \lambda \right)^{M-1} h = g_{M - 1} \quad \text{on} \quad \Gamma . \end{array} \right.
\]Corrigendum to: ``Integral equations method for solving a biharmonic inverse problem in detection of Robin coefficients''https://zbmath.org/1472.310062021-11-25T18:46:10.358925Z"Abdelhak, Hadj"https://zbmath.org/authors/?q=ai:hadj.abdelhak"Saker, Hacene"https://zbmath.org/authors/?q=ai:saker.haceneA typo in the authors' paper [ibid. 160, 436--450 (2021; Zbl 1459.31002)] is corrected.A proof of the Khavinson conjecturehttps://zbmath.org/1472.310072021-11-25T18:46:10.358925Z"Liu, Congwen"https://zbmath.org/authors/?q=ai:liu.congwenThe author gives a complete proof of the validity of the Khavinson conjecture. In order to state the conjecture, let \(h^\infty\) be the space of bounded harmonic functions on the unit ball \(\mathbb{B}^n\) of \(\mathbb{R}^n\), with \(n \geq 3\). For \(x \in \mathbb{B}^n\) we denote by \(C(x)\) the smallest number such that
\[
|\nabla u(x)| \leq C(x)\sup_{y \in \mathbb{B}^n}|u(y)|
\]
for all \(u \in h^\infty\). Similarly, for \(x\in \mathbb{B}^n\) and \(l\in \partial \mathbb{B}^n\), we denote by \(C(x,l)\) the smallest number such that
\[
|\langle\nabla u(x),l \rangle | \leq C(x,l)\sup_{y \in \mathbb{B}^n}|u(y)|
\]
for all \(u \in h^\infty\). As it is well known, both constants are finite. The Khavinson conjecture states that for \(x \in \mathbb{B}^n \setminus \{0\}\) we have
\[
C(x)=C\left(x,\frac{x}{|x|}\right)\, .
\]
The author shows the validity of the conjecture, by considering an equivalent optimization problem and by solving such a problem in terms of the Gegenbauer polynomials.Duality between range and no-response tests and its application for inverse problemshttps://zbmath.org/1472.310102021-11-25T18:46:10.358925Z"Lin, Yi-Hsuan"https://zbmath.org/authors/?q=ai:lin.yi-hsuan"Nakamura, Gen"https://zbmath.org/authors/?q=ai:nakamura.gen"Potthast, Roland"https://zbmath.org/authors/?q=ai:potthast.roland-w-e"Wang, Haibing"https://zbmath.org/authors/?q=ai:wang.haibingThe authors show the duality between range and no-response tests for an inverse boundary value problem for the Laplace equation in \(\Omega \setminus \overline{D}\) with an unknown obstacle \(D\) whose closure is contained in \(\Omega\). They consider the boundary value problem
\[
\left\{ \begin{array}{ll} \Delta u=0 & \mbox{in}\ \Omega \setminus \overline{D}\, ,\\
u=0 & \mbox{on}\ \partial D \, ,\\
u=f & \mbox{on}\ \partial \Omega \, . \end{array} \right.
\]
The Cauchy data is the pair made by Dirichlet datum \(f\) and the normal derivative \(\partial_\nu u_{|\partial \Omega}\). The inverse problem consists into identifying the unknown obstacle \(D\), knowing the Cauchy data \(\{f, \partial_\nu u_{|\partial \Omega}\}\).
The authors prove that there is a duality between the range test (RT) and the no-response test (NRT) for the inverse boundary value problem. As an application, they show that either using the RT or NRT, we can reconstruct the obstacle \(D\) from the Cauchy data if the solution \(u\) does not have any analytic extension across \(\partial D\).Corrigendum to: ``Viscosity solutions to degenerate complex Monge-Ampère equations''https://zbmath.org/1472.320202021-11-25T18:46:10.358925Z"Eyssidieux, Philippe"https://zbmath.org/authors/?q=ai:eyssidieux.philippe"Guedj, Vincent"https://zbmath.org/authors/?q=ai:guedj.vincent"Zeriahi, Ahmed"https://zbmath.org/authors/?q=ai:zeriahi.ahmedSummary: The proof of the comparison principle in our article [ibid. 64, No. 8, 1059--1094 (2011; Zbl 1227.32042)] is not complete. We provide here an alternative proof, valid in the ample locus of any big cohomology class, and discuss the resulting modifications.Parabolic complex Monge-Ampère equations on compact Kähler manifoldshttps://zbmath.org/1472.320212021-11-25T18:46:10.358925Z"Picard, Sebastien"https://zbmath.org/authors/?q=ai:picard.sebastien"Zhang, Xiangwen"https://zbmath.org/authors/?q=ai:zhang.xiangwenThe authors consider the parabolic Monge-Ampère equation
\[
\partial_t u =F(e^{-f}\det (\delta^i_j + \nabla^i\nabla_j u)),
\]
with
\[
\omega + i \partial \bar \partial u(x,t)>0,
\]
where \((X, \omega)\) is a compact Kähler manifold, \(f\in C^{\infty}(X, \mathbb{R})\) is a given function, \(F: \mathbb{R}_+ \rightarrow \mathbb{R}\) is a smooth strictly increasing function, and \(u\) is the unkown solution. Under the smooth initial data \(u(x,0)=u_0(x)\), the long-time existence and convergence of the parabolic equation are studied.
The main theorem (Theorem 1.1) is proved without any concavity (or convexity) assumption on the speed function \(F\), which includes the Kähler-Ricci flow (\(F(\rho)=\log \rho\)), the inverse Monge-Ampère flow (\(F(\rho)=1-\rho^{-1}\)), conformally Kähler Anomaly flow (\(F(\rho)=\rho\)), and the modified version of the Anomaly flow with zero slope parameter (\(F(\rho)=\rho^a\)) as special cases.
For the entire collection see [Zbl 1454.00057].Applications of a duality between generalized trigonometric and hyperbolic functionshttps://zbmath.org/1472.330022021-11-25T18:46:10.358925Z"Miyakawa, Hiroki"https://zbmath.org/authors/?q=ai:miyakawa.hiroki"Takeuchi, Shingo"https://zbmath.org/authors/?q=ai:takeuchi.shingoRelations between the trigonometric functions \(\sin\), \(\cos\), \(\tan\) and their hyperbolic counterparts \(\sinh\), \(\cosh\), \(\tanh\) are well known. In this paper, the authors consider generalized trigonometric and hyperbolic functions, defined via integral formulas and depending on two real parameters \(p\) and \(q\). These generalizations appear as solutions in certain differential equations and have been studied in the past. The main point of this article is that a certain extension of the admissible range of parameters \(p\) and \(q\) allows to express generalized hyperbolic sine and cosine in terms of their trigonometric counterparts and vice versa. This viewpoint allows a more or less automatic translations of equalities and inequalities from trigonometric to hyperbolic. Examples include double and multiple angle formulas as well as Mitrinović-Adamović-type inequalities.
Throughout most of the article, generalized trigonometric and hyperbolic tangent functions are ignored. Section~4 suggests that the lack of results in the trigonometric case might be due to the ``wrong'' definition of generalized tangent as mere quotient of sine and cosine. The authors suggest an alternative definition that recovers the usual tangent function but might allow a more natural generalization of relations known from ordinary trigonometric functions.
Some but not all of the formulas in this paper are new (the the authors meticulously provide references for what can be found in literature). Its main contribution is not a bunch of novel individual equalities and inequalities but is a concept two translate between generalized trigonometric and generalized hyperbolic functions.Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implementedhttps://zbmath.org/1472.340042021-11-25T18:46:10.358925Z"Bérard, Pierre"https://zbmath.org/authors/?q=ai:berard.pierre-h"Helffer, Bernard"https://zbmath.org/authors/?q=ai:helffer.bernardThis paper gives an account of results emanating from Sturm's seminal work on eigenvalues and eigenfunctions of the Dirichlet problem
\[
-y''(x)+q(x)y(x) = \lambda y(x)\text{ in }]0,1[,\ y(0)=y(1)=0.
\]
The authors are particularly interested in the statement that the number of the zeros of nontrivial real linear combinations of the first \(n\) real eigenfunctions is bounded by \(n-1\), where \(n\) is any positive integer. They present different proofs, interwoven with detailed accounts of the history of the presented approaches. This historical overview also briefly discusses the generalization to the Laplace operator \(-\Delta \) with Dirichlet boundary conditions on a bounded domain in \(\mathbb{R}^n\) and counterexamples thereof.
First, the authors present Liouville's proof. Then they give an alternative proof using the strategy proposed by Gelfand, considering the first eigenfunction of the \(n\)-particle Hamiltonian restricted to Fermions. This apprears to be the first complete proof following Gelfand's suggestion. Finally, Gelfand's strategy is compared with Kellog's approach and the resulting approach via oscillation kernels.
This paper is highly recommended to anyone interested in the history of Stum-Liouville theory or working in oscillation theory.Travelling wave solutions of the general regularized long wave equationhttps://zbmath.org/1472.340062021-11-25T18:46:10.358925Z"Zheng, Hang"https://zbmath.org/authors/?q=ai:zheng.hang"Xia, Yonghui"https://zbmath.org/authors/?q=ai:xia.yonghui"Bai, Yuzhen"https://zbmath.org/authors/?q=ai:bai.yuzhen"Wu, Luoyi"https://zbmath.org/authors/?q=ai:wu.luoyiThis paper studies the model of the general regularized long wave (GRLW) equation. The main contribution of this paper is to find that GRLW equation has extra kink and anti-kink wave solutions when $p = 2n + 1$, while it's not for $p = 2n$. The authors give the phase diagram and obtained possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits.Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equationhttps://zbmath.org/1472.340752021-11-25T18:46:10.358925Z"Zhang, Lijun"https://zbmath.org/authors/?q=ai:zhang.lijun"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.6"Khalique, Chaudry Masood"https://zbmath.org/authors/?q=ai:khalique.chaudry-masood"Bai, Yuzhen"https://zbmath.org/authors/?q=ai:bai.yuzhenThis paper studies the model of the generalized Camassa-Holm-Novikov (gCHN) equation. The main contribution of this paper is to discuss the singular wave solutions including peakons and cuspons. They give the phase diagram and obtain the exact explicit parametric representation of the traveling wave solutions corresponding to these singular wave solutions.Mean-field and graph limits for collective dynamics models with time-varying weightshttps://zbmath.org/1472.340842021-11-25T18:46:10.358925Z"Ayi, Nathalie"https://zbmath.org/authors/?q=ai:ayi.nathalie"Pouradier Duteil, Nastassia"https://zbmath.org/authors/?q=ai:duteil.nastassia-pouradierSummary: In this paper, we study a model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinions' evolution. We explore the natural question of the large population limit with two approaches: the now classical mean-field limit and the more recent graph limit. After establishing the existence and uniqueness of solutions to the models that we will consider, we provide a rigorous mathematical justification for taking the graph limit in a general context. Then, establishing the key notion of \textit{indistinguishability}, which is a necessary framework to consider the mean-field limit, we prove the subordination of the mean-field limit to the graph one in that context. This actually provides an alternative proof for the mean-field limit. We conclude by showing some numerical simulations to illustrate our results.Stochastic approaches to Lagrangian coherent structureshttps://zbmath.org/1472.341112021-11-25T18:46:10.358925Z"Balasuriya, Sanjeeva"https://zbmath.org/authors/?q=ai:balasuriya.sanjeevaSummary: This note discusses a connection between deterministic Lagrangian coherent structures (robust fluid parcels which move coherently in unsteady fluid flows according to a deterministic ordinary differential equation), and the incorporation of noise or stochasticity which leads to the Fokker-Planck equation (a partial differential equation governing a probability density function). The link between these is via a stochastic ordinary differential equation. It is argued that a closer investigation of the stochastic differential equation offers additional insights to both the other approaches, and in particular to uncertainty quantification in Lagrangian coherent structures.
For the entire collection see [Zbl 1462.35005].An introduction to partial differential equations (with Maple). A concise coursehttps://zbmath.org/1472.350012021-11-25T18:46:10.358925Z"Li, Zhilin"https://zbmath.org/authors/?q=ai:li.zhilin"Norris, Larry"https://zbmath.org/authors/?q=ai:norris.larry-kPublisher's description: The book is designed for undergraduate or beginning level graduate students, and students from interdisciplinary areas including engineers, and others who need to use partial differential equations, Fourier series, Fourier and Laplace transforms. The prerequisite is a basic knowledge of calculus, linear algebra, and ordinary differential equations.
The textbook aims to be practical, elementary, and reasonably rigorous; the book is concise in that it describes fundamental solution techniques for first order, second order, linear partial differential equations for general solutions, fundamental solutions, solution to Cauchy (initial value) problems, and boundary value problems for different PDEs in one and two dimensions, and different coordinates systems. Analytic solutions to boundary value problems are based on Sturm-Liouville eigenvalue problems and series solutions.
The book is accompanied with enough well tested Maple files and some Matlab codes that are available online. The use of Maple makes the complicated series solution simple, interactive, and visible. These features distinguish the book from other textbooks available in the related area.Partial differential equations. Classical theory with a modern touchhttps://zbmath.org/1472.350022021-11-25T18:46:10.358925Z"Nandakumaran, A. K."https://zbmath.org/authors/?q=ai:nandakumaran.akambadath-keerthiyil"Datti, P. S."https://zbmath.org/authors/?q=ai:datti.p-sThis textbook presents a nice and smooth introduction to the study of partial differential equations (PDEs). The book is divided into 12 chapters as follows. Chapter 1 contains a short introduction which motivates the study of PDEs. Chapter 2 contains basic preliminaries such as relevant topics from multivariable calculus, first order ODEs and Fourier transform. Chapter 3 discusses first order PDEs, focussing on the method of characteristics. It contains both linear and nonlinear equations in several variables. Chapter 4 is devoted to the Hamilton-Jacobi equation along with Legendre transformation. Chapter 5 presents an introduction to conservation laws.
The following chapters discuss second order PDEs. More precisely the authors include the classification of second order linear PDEs (Chapter 6) followed by Laplace and Poisson equation (Chapter 7), heat equation (Chapter 8), 1D and higher dimensional wave equation (Chapters 9 and 10). Chapter 11 discusses the Cauchy-Kovalevsky theorem and its generalizations, while the last chapter contains a presentation of Sobolev spaces. Each chapter concludes with some historical notes and motivation for further reading as well as a good number of relevant exercises.
The work under discussion is a nice presentation of PDEs and proves to be a valuable source for undergraduate and graduate students at all levels. It highlights the importance of studying differential equations both in the setting of classical solutions as well as weak solutions.Nonlinear fractional Schrödinger equations in \(\mathbb R^N\)https://zbmath.org/1472.350032021-11-25T18:46:10.358925Z"Ambrosio, Vincenzo"https://zbmath.org/authors/?q=ai:ambrosio.vincenzoThis monograph is concerned with, broadly speaking, nonlinear differential equations in the whole Euclidean space \(\mathbb{R}^N\) governed by a fractional differential operator. As a model, one may consider the equation \((-\Delta)^s u + V(x)u = f(x,u)\), with \(0<s<1\), under suitable assumptions on the potential \(V\) and on the nonlinearity \(f\).
The book is divided into seventeen chapters ranging from functional-analytic preliminaries to very specific types of fractional equations like fractional Schrödinger-Poisson equations, equations with an external magnetic field, equations with critical or sub-critical growth, and much more. Many results were proved by the authors and his collaborators in the last years, and several proofs have been simplified or improved in comparison with the original ones.
The text offers a thourough overview of fractional Schrödinger equations in \(\mathbb{R}^N\) that can be used for a course at a graduate or Ph.D. level. A rich bibliography of more than three hundred references completes the book. Both researchers who approach the subject and experts looking for a solid reference will appreciate this volume.Asymptotic theory of dynamic boundary value problems in irregular domainshttps://zbmath.org/1472.350042021-11-25T18:46:10.358925Z"Korikov, Dmitrii"https://zbmath.org/authors/?q=ai:korikov.dmitrii-v"Sarafanov, Oleg"https://zbmath.org/authors/?q=ai:sarafanov.oleg-v"Sarafanov, Oleg"https://zbmath.org/authors/?q=ai:sarafanov.oleg-vThis book concerns so-called dynamic boundary value problems, i.e. linear PDEs of the type
\[
\mathcal{L}(x,D_x,D_t)u(x,t)=f(x,t), \; (x,t) \in G \times \mathbb R
\]
with certain boundary conditions on the lateral boundary \(\partial G \times \mathbb R\). The spatial domain \(G\) is supposed to be irregular: It may have edges of various dimensions on its boundary \(\partial G\) (like polygons, polyhedra, cones, wedges or lenses). Or it may depend on a small parameter \(\varepsilon \ge 0\) such that for \(\varepsilon =0\) there exist edges on \(\partial G\) and/or singular points in \(G\) and that, in the transition from \(\varepsilon=0\) to \(\varepsilon>0\), the edges are smoothed and/or the singular points become small cavities.
The goal is to state results about existence and uniqueness of strong solutions and to describe the asymptotics of the solutions near the edges and/or for \(\varepsilon \to 0\).
The considered PDEs are the wave equation, general hyperbolic systems of second-order PDEs, systems of elastodynamics, the Maxwell system, the Schrödinger equation and the Germain-Lagrage plate equation. The boundary conditions are of homogeneous Dirichlet or Neumann type, of ideal conductivity type (for the Maxwell system) and of clamped boundary type (for the plate equation).
All problems are handled by means of an universal approach called ``weighted combined estimates''. These estimates are ``combined'' because they involve derivatives of different orders of the solution in the ``elliptic'' zone (close to the vertex) and in the ``hyperbolic'' zone (far from the vertex), respectively. The estimates are obtained by applying the Fourier transform to the original problem, getting a family of problems in a cone, proving weighted combined estimates in the cone and, finally, applying the inverse Fourier transform (which is possible because the estimates are uniform with respect to the family parameters).
Roughly speaking, the solutions are shown to have asymptotic expansions of the type
\[
u(x,t) \sim \sum_{j\ge 0} c_j(t)u_j(x,t).
\]
The larger \(j\), the smaller (near the singularity) are the normalized wave forms \(u_j(x,t)\), and the wave intensities \(c_j(t)\) are determined by the past of the evolution.Nonlinear problems with lack of compactnesshttps://zbmath.org/1472.350052021-11-25T18:46:10.358925Z"Molica Bisci, Giovanni"https://zbmath.org/authors/?q=ai:molica-bisci.giovanni"Pucci, Patrizia"https://zbmath.org/authors/?q=ai:pucci.patriziaThis book presents recent research results on nonlinear problems with lack of compactness via critical point theory, obtained by several mathematicians. The topics covered include several nonlinear problems in the Euclidean setting as well as variational problems on manifolds. It is divided into three parts. In the first part of the book, the authors deal with the existence of solutions for quasilinear elliptic equations in \(\mathbb{R}^{N}\), involving general operators with nonstandard growth, as well as critical nonlinearities.
In the second part of the book, the existence of multiple solutions has been treated via a group-theoretical invariance in the Hilbertian framework for different problems such as the one-parameter critical elliptic equation in \(\mathbb{R}^{N}\), the scalar field equation settled on a strip-like domain of the Euclidean space \(\mathbb{R}^{N}\) and elliptic equations on the unit sphere \(S^{N}\rightarrow\mathbb{R}^{N+1}\), with \(N\geq 2\), endowed by the induced Riemannian metric and involving a possibly critical nonlinear term. In these problems, the settings are responsible for the loss of compactness.
The third part of the book is dedicated to non-compact problems arising from geometry. More precisely, the authors deal with subelliptic problems on Carnot groups and treat elliptic problems on homogeneous Hadamard manifolds, i.e. Riemannian manifolds which are complete, simply connected, with everywhere non-positive sectional curvature and with a transitive group isometries. Moreover, by taking the advantage of the intrinsic nature of the hyperbolic geometry, elliptic problems on the Poincaré ball model are studied. Finally, the authors give a celebrated principle of symmetric criticality of R. Palais intensively used along Parts II and III of the book.Semilinear elliptic equations. Classical and modern theorieshttps://zbmath.org/1472.350062021-11-25T18:46:10.358925Z"Suzuki, Takashi"https://zbmath.org/authors/?q=ai:suzuki.takashiThis book presents in detail both classical and modern methods of study semilinear elliptic equations, i.e. methods of studying qualitative properties of solutions using variational techniques, the maximum principle, blowup analysis, spectral theory, topological methods, etc.
The book is divided into two parts.
Part I is devoted to the fundamental theory and consists of four chapters.
The topic of the first chapter is the existence of the solutions. The first issue is the notion of the critical exponent, which divides the problem into existence and nonexistence alternatives. Method of variations are used to establish the existence of the solution for the power nonlinearity. Three ways to tackle the problem are presented: Pohozaev identity, Lagrangian multiplier, critical exponent, mountain pass, and the Nehari method. The proof of Brezis-Kato's theorem is included for completeness.
In Chapter 2 two methods are presented to solve the problem of uniqueness: introducing parameters and utilizing the symmetry. It starts with a discussion of the ignition model as an introduction, after which the structure of its stationary state is revealed. The author has added further results on the blowup of the nonstationary solution. The ODE approach is then justified by the symmetry result of Gidas, Ni, and Nirenberg theorem. Then the strong maximum principle is applied to ensure the convexity of levelsets of the solution to the heat equation. The last section is devoted to the symmetric criticality, where an efficient use of the mountain pas lemma of Ghouss-Preiss is applied to confirm the emergence of nonradial solutions in any mode. Then the Gel'fand equation appears.
Chapter 3 begins with the theory of surfaces associated with the complex structure, and observes remarkable structures of the Boltzmann-Poisson equation. Then the complex function theory is applied to classify the singular limits of the Boltzmann-Poisson equation and to construct one-point blow-up solutions. The next sections deal with the applications of the complex structure (Wenston's theory, Moseley-Wente's theory).
The first section of Chapter 4 describes the classical theorem of Faber-Krahn, the isoperimetric inequality for the first eigenvalue of the Laplacian under the Dirichlet boundary condition, based on the method of rearrangement. The second section describes the equimeasurable transformation and its applications. The potential theory for spherically harmonic functions is developed as in the Harnack principle, via Nehari's isoperimetric inequality in the third section. The analytic proof of Hamilton's result on the normalized Ricci flow is provided in the last section.
Part II is concerned with the applications oriented towards extremal structures of several specific models.
First, Chapter 5 is concerned with the supplementary topics related to Part I. In the first and second section the author shows an alternative method of variations to the mountain pass lemma with application to the construction of iterative sequences, and several problems resolved in accordance with the method of the moving plane. The third section is devoted to the method of scaling applied to the Boltzmann-Poisson equation and its higher-dimensional version. In the fourth section the author studies a blow-up pattern emerging form the differential inequality \(-\Delta u \le f(u)\) by the method of equimeasurable transformations. In the next section author deals with the Trudinger-Moser inequality in two dimensions in the strong form. The final section is devoted to the topological deformation of level sets of functionals associated with the sinh-Poisson equations in the flavor of the Morse theory.
In Chapter 6 the author comes back to the classical study of the Gel'fand equation on an annulus. He describes the profile of nonradially symmetric solutions in 2D case, based on the variational and bifurcation analysis in accordance with the Lioville integral. Then he investigates the structure of the set of radially symmetric solutions in higher-dimension.
The last chapter is devoted to the regularity of the solutions and Hardy-BMO duality, respectively. In the first section the author describes Moser's iteration scheme applied to \(p\)-harmonic functions. This scheme arises here with the BMO estimate. Then, Hardy spaces emerge with the duality, and a regularity theorem is obtained. The last two sections are devoted to the formation of bubbles made by harmonic maps and Smoluchowski-Poisson equation.On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equationshttps://zbmath.org/1472.350072021-11-25T18:46:10.358925Z"Johansson, B. Tomas"https://zbmath.org/authors/?q=ai:johansson.b-tomasSummary: Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.Some non-linear systems of PDEs related to inverse problems in conductivityhttps://zbmath.org/1472.350082021-11-25T18:46:10.358925Z"Maestre, Faustino"https://zbmath.org/authors/?q=ai:maestre.faustino"Pedregal, Pablo"https://zbmath.org/authors/?q=ai:pedregal.pabloThe main focus of the article is the Euler-Lagrange system associated with the functional
\[
I(u) = \int_{\Omega} |\nabla u_1| |\nabla u_2| \mathrm{d} x.
\]
This gives a system of PDEs with a connection to the Calderón's problem: the conductivity will be
\[
\gamma = \frac{\|\nabla u_2\|}{\|\nabla u_1\|}.
\]
The functional is non-coercive and not quasiconvex, foiling the direct method in calculus of variations. The authors present a modified problem with is non-negative and zero precisely at solutions and consider its quasiconvexification. This problem has a connection to a boundary condition for \(u_1\) and \(u_2\).
Numerical tests investigate the functioning of Newton-Raphson method for solving the nonlinear system of equations.He-Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamicshttps://zbmath.org/1472.350092021-11-25T18:46:10.358925Z"Nadeem, Muhammad"https://zbmath.org/authors/?q=ai:nadeem.muhammad|nadeem.muhammad-faisal"He, Ji-Huan"https://zbmath.org/authors/?q=ai:he.ji-huan|he.jihuanSummary: In this article, we suggest an alternative approach for the study of some partial differential equations (PDEs) arising in physical phenomena such as chemical kinetics and population dynamics. He-Laplace variational iteration method (He-LVIM) is a simple but effective way where the variational iteration method (VIM) is coupled with Laplace transforms method. He's polynomials are obtained by the homotopy perturbation method (HPM) to deal with the nonlinear terms. Fisher equation, the generalized Fisher equation and the nonlinear diffusion equation of the Fisher type are suggested to demonstrate the accuracy and stability of this method.Localized structures on librational and rotational travelling waves in the sine-Gordon equationhttps://zbmath.org/1472.350102021-11-25T18:46:10.358925Z"Pelinovsky, Dmitry E."https://zbmath.org/authors/?q=ai:pelinovsky.dmitry-e"White, Robert E."https://zbmath.org/authors/?q=ai:white.robert-eSummary: We derive exact solutions to the sine-Gordon equation describing localized structures on the background of librational and rotational travelling waves. In the case of librational waves, the exact solution represents a localized spike in space-time coordinates (a rogue wave) that decays to the periodic background algebraically fast. In the case of rotational waves, the exact solution represents a kink propagating on the periodic background and decaying algebraically in the transverse direction to its propagation. These solutions model the universal patterns in the dynamics of fluxon condensates in the semi-classical limit. The different dynamics are related to modulational instability of the librational waves and modulational stability of the rotational waves.Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuationhttps://zbmath.org/1472.350112021-11-25T18:46:10.358925Z"Banerjee, Agnid"https://zbmath.org/authors/?q=ai:banerjee.agnid"Manna, Ramesh"https://zbmath.org/authors/?q=ai:manna.rameshSummary: In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with \textit{N. Garofalo} in [``Carleman estimates for Baouendi-Grushin operators with applications to quantitative uniqueness and strong unique continuation'', Preprint, \url{arXiv:1903.08382}] where similar estimates were established for the ``constant coefficient'' Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.On critical dipoles in dimensions \(n \geq 3\)https://zbmath.org/1472.350122021-11-25T18:46:10.358925Z"Blake Allan, S."https://zbmath.org/authors/?q=ai:blake-allan.s"Gesztesy, Fritz"https://zbmath.org/authors/?q=ai:gesztesy.fritzSummary: We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials \(V_\gamma(x) = \gamma(u, x) | x |^{- 3}\), \(x \in \mathbb{R}^n\setminus\{0 \}\), \(\gamma \in [0, \infty)\), \(u \in \mathbb{R}^n\), \(| u | = 1\), \(n \in \mathbb{N}\), \(n \geqslant 3\). More precisely, for \(n \geqslant 3\), we provide an alternative proof of the existence of a critical dipole coupling constant \(\gamma_{c , n} > 0\), such that
\begin{align*}
&\text{for all } \gamma \in [ 0 , \gamma_{c , n} ] \text{, and all } u \in \mathbb{R}^n,\ | u | = 1 , \\
&\quad\int_{\mathbb{R}^n} d^n x |({\nabla} f)(x) |^2 \geqslant \pm \gamma \int_{\mathbb{R}^n} d^n x(u, x) | x |^{- 3} | f(x) |^2,\quad f \in D^1( \mathbb{R}^n)
\end{align*}
with \(D^1( \mathbb{R}^n)\) denoting the completion of \(C_0^\infty( \mathbb{R}^n)\) with respect to the norm induced by the gradient. Here \(\gamma_{c , n}\) is sharp, that is, the largest possible such constant. Moreover, we discuss upper and lower bounds for \(\gamma_{c , n} > 0\) and develop a numerical scheme for approximating \(\gamma_{c , n} \).
This quadratic form inequality will be a consequence of the fact
\[\overline{\left[ - {\Delta} + \gamma ( u , x ) | x |^{- 3} \right]\! |_{C_0^\infty ( \mathbb{R}^n \setminus \{ 0 \} )}} \geqslant 0 \text{ if and only if } 0 \leqslant \gamma \leqslant \gamma_{c , n}\]
in \(L^2( \mathbb{R}^n)\) (with \(\overline{T}\) the operator closure of the linear operator \(T\)).
We also consider the case of multicenter dipole interactions with dipoles centered on an infinite discrete set.On a class of sharp multiplicative Hardy inequalitieshttps://zbmath.org/1472.350132021-11-25T18:46:10.358925Z"Guzu, D."https://zbmath.org/authors/?q=ai:guzu.dorian"Hoffmann-Ostenhof, T."https://zbmath.org/authors/?q=ai:hoffmann-ostenhof.thomas"Laptev, A."https://zbmath.org/authors/?q=ai:laptev.ariSummary: A class of weighted Hardy inequalities is treated. The sharp constants depend on the lowest eigenvalues of auxiliary Schrödinger operators on a sphere. In particular, for some block radial weights these sharp constants are given in terms of the lowest eigenvalue of a Legendre type equation.Sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fieldshttps://zbmath.org/1472.350142021-11-25T18:46:10.358925Z"Hamamoto, Naoki"https://zbmath.org/authors/?q=ai:hamamoto.naoki"Takahashi, Futoshi"https://zbmath.org/authors/?q=ai:takahashi.futoshiThe authors proved sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields. Namely, let \(\gamma\not=1-N/2\) be a real number and let \(u\in C^{\infty}_{c}(\mathbb{R}^{N})^{N}\) be a curl-free vector field. Then, there are optimal constants \(H_{N,\gamma}\) and \(R_{N,\gamma}\) (given explicitly) satisfying:
\begin{itemize}
\item [\(i)\)] If \(u(0)=0\) and \(\gamma<1-N/2\), then
\[
H_{N,\gamma}\int_{\mathbb{R}^{N}}\frac{|u|^2}{|x|^2}|x|^{2\gamma}dx\le \int_{\mathbb{R}^{N}}|\nabla u|^{2}|x|^{2\gamma}dx.
\]
\item [\(ii)\)] If \(\int_{\mathbb{R}^N}|x|^{2\gamma-4}|u|^{2}dx<\infty\), then
\[
R_{N,\gamma}\int_{\mathbb{R}^{N}}\frac{|u|^2}{|x|^4}|x|^{2\gamma}dx\le \int_{\mathbb{R}^{N}}|\Delta u|^{2}|x|^{2\gamma}dx.
\]
\end{itemize}
These results complement the former work by [\textit{O. Costin} and \textit{V. Maz'ya}, Calc. Var. Partial Differ. Equ. 32, No. 4, 523--532 (2008; Zbl 1147.35122)] on the sharp Hardy-Leray inequality for axisymmetric divergence-free vector fields.\(L^p\)-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with \(p\)-integrable exterior derivativehttps://zbmath.org/1472.350152021-11-25T18:46:10.358925Z"Lewintan, Peter"https://zbmath.org/authors/?q=ai:lewintan.peter"Neff, Patrizio"https://zbmath.org/authors/?q=ai:neff.patrizioSummary: For \(n\geq 2\) and \(1<p<\infty\) we prove an \(L^p\)-version of the generalized Korn-type inequality for incompatible, \(p\)-integrable tensor fields \(P:\Omega\rightarrow\mathbb{R}^{n\times n}\) having \(p\)-integrable generalized \(\underline{\mathrm{Curl}}\) and generalized vanishing tangential trace \(P\tau_l=0\) on \(\partial\Omega\), denoting by \(\{\tau_l\}_{l=1,\dots,n-1}\) a moving tangent frame on \(\partial\Omega\), more precisely we have:
\[
\Vert P\Vert_{L^p(\Omega,\mathbb{R}^{n\times n})}\leq c\left(\left\Vert \operatorname{sym}P\right\Vert_{L^p(\Omega,\mathbb{R}^{n\times n})}+\left\Vert\underline{\mathrm{Curl}}\,P \right\Vert_{L^p(\Omega,(\mathfrak{so}(n))^n)}\right),
\]
where the generalized \(\underline{\mathrm{Curl}}\) is given by \((\underline{\mathrm{Curl}}\,P)_{ijk}:=\partial_iP_{kj}-\partial_jP_{ki}\) and \(c=c(n,p,\Omega )>0\).A note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalitieshttps://zbmath.org/1472.350162021-11-25T18:46:10.358925Z"Martínez-Perales, Javier C."https://zbmath.org/authors/?q=ai:martinez-perales.javier-cSummary: The main result of this paper supports a conjecture by Pérez and Rela about the properties of the weight appearing in their recent self-improving result of generalized inequalities of Poincaré-type in the Euclidean space. The result we obtain does not need any condition on the weight, but still is not fully satisfactory, even though the result by Pérez and Rela is obtained as a corollary of ours. Also, we extend the conclusions of their theorem to the range \(p<1\).
As an application of our result, we give a unified vision of weighted improved Poincaré-type inequalities in the Euclidean setting, which gathers both weighted improved classical and fractional Poincaré inequalities within an approach which avoids any representation formula. We obtain results related to some already existing results in the literature and furthermore we improve them in some aspects. Finally, we also explore analog inequalities in the context of metric spaces by means of the already known self-improving results in this setting.Symmetries of differential ideals and differential equationshttps://zbmath.org/1472.350172021-11-25T18:46:10.358925Z"Kaptsov, Oleg V."https://zbmath.org/authors/?q=ai:kaptsov.oleg-viktorovichSummary: The paper deals with differential rings and partial differential equations with coefficients in some algebra. We introduce symmetries and the conservation laws to the differential ideal of an arbitrary differential ring. We prove that a set of symmetries of an ideal forms a Lie ring and give a precise criterion when a differentiation is a symmetry of an ideal. These concepts are applied to partial differential equations.A posteriori error estimates for numerical solutions to hyperbolic conservation lawshttps://zbmath.org/1472.350182021-11-25T18:46:10.358925Z"Bressan, Alberto"https://zbmath.org/authors/?q=ai:bressan.alberto"Chiri, Maria Teresa"https://zbmath.org/authors/?q=ai:chiri.maria-teresa"Shen, Wen"https://zbmath.org/authors/?q=ai:shen.wenThe paper at hand studies a posteriori criteria for the convergence of numerical schemes for systems of hyperbolic conservation laws in one spatial dimension.
The results can be understood in the wider context of convergene of approximation schemes (not necessarily numerical methods but also, e.g., vanishing viscosity) for systems of hyperbolic conservation laws in one spatial dimension. Previous convergence results relied on compensated compactness and, thus, do not provide information on uniqueness of limits or convergence rates.
The paper at hand aims at overcoming these limitations by using the $L^1$ stability theory, developed mainly by the first author, to compare approximate solutions to the entropy solution semi-group. There is, however, a fundamental obstruction to proving a priori results in this setting, since the counterexample from [\textit{A. Bressan} et al., Commun. Pure Appl. Math. 59, No. 11, 1604--1638 (2006; Zbl 1122.35074)] shows that in certain cases standard schemes such as Godunov's method are unstable in total variation.
The authors consider a large class of numerical schemes satisfying the Lipschitz continuity of the approximate solution (in a suitable sense) and consistency with the weak forms of the equation and of the entropy inequality. They show that, if post-processed versions of (actually computed) numerical solutions to such schemes satisfy two criteria (that can be verified in an a posteriori sense), namely that the total variation (in space) of the approximate solution remains small at all times, outside some small strips in space-time the oscisllation of numerical solutions remains small then the $L^1$ error is small.Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equationshttps://zbmath.org/1472.350192021-11-25T18:46:10.358925Z"Boyko, Vyacheslav M."https://zbmath.org/authors/?q=ai:boyko.vyacheslav-m"Lokaziuk, Oleksandra V."https://zbmath.org/authors/?q=ai:lokaziuk.oleksandra-v"Popovych, Roman O."https://zbmath.org/authors/?q=ai:popovych.roman-oSummary: Essentially generalizing Lie's results, we prove that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein-Gordon equations is the first-order prolongation of its point equivalence groupoid, and then we carry out the complete group classification of this class. Since it is normalized, the algebraic method of group classification is naturally applied here. Using the specific structure of the equivalence group of the class, we essentially employ the classical Lie theorem on realizations of Lie algebras by vector fields on the line. This approach allows us to enhance previous results on Lie symmetries of equations from the class and substantially simplify the proof. After finding a number of integer characteristics of cases of Lie-symmetry extensions that are invariant under action of the equivalence group of the class under study, we exhaustively describe successive Lie-symmetry extensions within this class.The dispersionless Veselov-Novikov equation: symmetries, exact solutions, and conservation lawshttps://zbmath.org/1472.350202021-11-25T18:46:10.358925Z"Morozov, Oleg I."https://zbmath.org/authors/?q=ai:morozov.oleg-i"Chang, Jen-Hsu"https://zbmath.org/authors/?q=ai:chang.jen-hsuSummary: We study symmetries, invariant solutions, and conservation laws for the dispersionless Veselov-Novikov equation. The emphasis is placed on cases when the odes involved in description of the invariant solutions are integrable by quadratures. Then we find some non-invariant solutions, in particular, solutions that are polynomials of an arbitrary degree \(N \ge 3\) with respect to the spatial variables. Finally we compute all conservation laws that are associated to cosymmetries of second order.Propagation dynamics of Lotka-Volterra competition systems with asymmetric dispersal in periodic habitatshttps://zbmath.org/1472.350212021-11-25T18:46:10.358925Z"Hao, Yu-Xia"https://zbmath.org/authors/?q=ai:hao.yu-xia"Li, Wan-Tong"https://zbmath.org/authors/?q=ai:li.wan-tong"Wang, Jia-Bing"https://zbmath.org/authors/?q=ai:wang.jiabingSummary: This paper is mainly concerned with the new types of entire solutions which are different from pulsating fronts of Lotka-Volterra competition systems with asymmetric dispersal in spatially periodic habitats. The asymmetry of kernel function leads to a great influence on the profile of pulsating fronts and the sign of wave speeds, which gives rise to the properties of the entire solution more complex and diverse. We first give a relationship between the critical speeds \(c^\ast(\xi)\) and \(c^\ast(- \xi)\), corresponding to the minimal speeds of two pulsating fronts propagating in the direction of \(\xi\) and \(- \xi \), respectively. Then, the exponential behavior of pulsating fronts as they approach their limiting states is obtained. Finally, by considering the interactions of two different pulsating fronts coming from two opposite/same directions and applying the super-/subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of some different types of entire solutions defined for all time and whole space. Moreover, we give some numerical simulations to describe intuitively these obtained entire solutions.Periodic solutions to nonlinear Euler-Bernoulli beam equationshttps://zbmath.org/1472.350222021-11-25T18:46:10.358925Z"Chen, Bochao"https://zbmath.org/authors/?q=ai:chen.bochao"Gao, Yixian"https://zbmath.org/authors/?q=ai:gao.yixian"Li, Yong"https://zbmath.org/authors/?q=ai:li.yong.1This rather technical paper shows the existence of time periodic solutions for an Euler-Bernoulli beam equation with periodic source terms of periode \(2\pi\). The argument employed to prove this result combines a Lyapunov-Schmidt reduction to the Nash-Moser method.Periodic solutions to Klein-Gordon systems with linear couplingshttps://zbmath.org/1472.350232021-11-25T18:46:10.358925Z"Chen, Jianyi"https://zbmath.org/authors/?q=ai:chen.jianyi"Zhang, Zhitao"https://zbmath.org/authors/?q=ai:zhang.zhitao"Chang, Guijuan"https://zbmath.org/authors/?q=ai:chang.guijuan"Zhao, Jing"https://zbmath.org/authors/?q=ai:zhao.jing.2|zhao.jing|zhao.jing.1|zhao.jing.3Summary: In this paper, we study the nonlinear Klein-Gordon systems arising from relativistic physics and quantum field theories
\[\begin{cases}
u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)=0, \\
v_{tt}\,-v_{xx}+bv+\varepsilon u+g(t,x,v)\,=0,
\end{cases}\]
where \(u,v\) satisfy the Dirichlet boundary conditions on spatial interval \([0,\pi], b>0\) and \(f,g\) are \(2\pi\)-periodic in \({t}\). We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as \(\varepsilon\) goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on \({f}\) and \({g}\), we obtain the solutions \((u_{\varepsilon},v_{\varepsilon})\) with time period \(2\pi\) for the problem as the linear coupling constant \(\varepsilon\) is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as \(\varepsilon\to 0, (u_{\varepsilon},v_{\varepsilon})\) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.Layer dynamics for the one dimensional \(\varepsilon\)-dependent Cahn-Hilliard/Allen-Cahn equationhttps://zbmath.org/1472.350242021-11-25T18:46:10.358925Z"Antonopoulou, D. C."https://zbmath.org/authors/?q=ai:antonopoulou.dimitra-c"Karali, G."https://zbmath.org/authors/?q=ai:karali.georgia-d"Tzirakis, K."https://zbmath.org/authors/?q=ai:tzirakis.konstantinosSummary: We study the dynamics of the one-dimensional \(\varepsilon\)-dependent Cahn-Hilliard/Allen-Cahn equation within a neighborhood of an equilibrium of \(N\) transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a mass-conservation constraint in place of one of the zero-mass flux boundary conditions at \(x=1\). Motivated by the study of \textit{J. Carr} and \textit{R. L. Pego} on the layered metastable patterns of Allen-Cahn in [Commun. Pure Appl. Math. 42, No. 5, 523--576 (1989; Zbl 0685.35054)], and by this of \textit{P. W. Bates} and \textit{J. Xun} [J. Differ. Equations 111, No. 2, 421--457 (1994; Zbl 0805.35046)] for the Cahn-Hilliard equation, we implement an \(N\)-dimensional, and a mass-conservative \(N-1\)-dimensional manifold respectively; therein, a metastable state with \(N\) transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized Cahn-Hilliard/Allen-Cahn operator, and specify wide families of \(\varepsilon\)-dependent weights \(\delta(\varepsilon)\), \(\mu(\varepsilon)\), acting at each part of the operator, for which the dynamics are stable and rest exponentially small in \(\varepsilon\). Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to Allen-Cahn, or, when the mass is conserved, close to the Cahn-Hilliard solution.An Allen-Cahn equation based on an unconstrained order parameter and its Cahn-Hilliard limithttps://zbmath.org/1472.350252021-11-25T18:46:10.358925Z"Miranville, Alain"https://zbmath.org/authors/?q=ai:miranville.alain-mSummary: Our aim in this paper is to study an Allen-Cahn model based on a microforce balance and an unconstrained order parameter. We obtain the existence, uniqueness and regularity of solutions and prove that the solutions converge to those to the original Cahn-Hilliard equation on finite time intervals as a small parameter goes to zero.Weakly nonlinear analysis of peanut-shaped deformations for localized spots of singularly perturbed reaction-diffusion systemshttps://zbmath.org/1472.350262021-11-25T18:46:10.358925Z"Wong, Tony"https://zbmath.org/authors/?q=ai:wong.tony-w-h|wong.tony-siu-tung"Ward, Michael J."https://zbmath.org/authors/?q=ai:ward.michael-jHomogenization of random convolution energieshttps://zbmath.org/1472.350272021-11-25T18:46:10.358925Z"Braides, Andrea"https://zbmath.org/authors/?q=ai:braides.andrea"Piatnitski, Andrey"https://zbmath.org/authors/?q=ai:piatnitski.andrey-lSummary: We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity, we prove that the \(\Gamma\)-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, which can be extended to this `asymptotically local' case. As a particular application, we derive a homogenization theorem on random perforated domains.Rigorous derivation of a mean field model for the Ostwald ripening of thin filmshttps://zbmath.org/1472.350282021-11-25T18:46:10.358925Z"Dai, Shibin"https://zbmath.org/authors/?q=ai:dai.shibinDuring the late stages of the evolution of thin liquid films on a solid substrate, liquid droplets are connected by an ultra-thin residual film. Their number decreases due to migration and collision on the one hand, and exchange of matter through a diffusive field in the residual film on the other hand. Supposing that, at time \(t>0\), there are \(N(t)\ge 1\) droplets \(\{B(x_i,R_i(t))\ :\ 1\le i \le N(t)\}\) in the square \(\Omega_{\mathcal{L}} =(0,\mathcal{L})^2\) and neglecting the motion of the centers \(x_i\) due to the no-slip boundary condition for the fluid at the substrate, the dynamics of the radii \((R_i)\) and the diffusive field \(u\) may be described by
\begin{align*}
- \Delta u(t,x) &= 0, \qquad x\in \Omega_{\mathcal{L}}\setminus \bigcup_{i=1}^{N(t)} \bar{B}(x_i,R_i(t)), \ t>0, \\
u(t,x) &= \frac{1}{R_i(t)}, \qquad x\in B(x_i,R_i(t)), \ t>0, \\
\frac{dR_i}{dt}(t) &= \frac{1}{R_i(t)^2} \int_{\partial B(x_i,R_i(t))} [\nabla u(t,s)\cdot \mathbf{n}(s)]\ ds, \qquad t>0,
\end{align*}
supplemented with periodic boundary conditions on \(\partial\Omega_{\mathcal{L}}\). In the above integral term, \([\nabla u(t,s)\cdot \mathbf{n}(s)]\) denotes the jump of the normal gradient of \(u\) across the boundary of \(B(x_i,R_i(t))\). After introducing a small parameter \(\varepsilon>0\) and scaling \(\mathcal{L}\), \(x\), \(t\), \(N\), \((R_i)\), and \(u\) in an appropriate way, homogeneization techniques are used to establish the convergence of the rescaled diffusion fields to a mean field \(u_*\). The latter is a weak solution to
\begin{align*}
-\Delta u_*(t,y) + 2\pi\delta \int_0^\infty \left( u_*(t,y) - \frac{1}{r} \right) f(t,y,r)\ dr & = 0, \qquad r\in (0,\infty),\\
\partial_t f(t,y,r) + \partial_r \left( \frac{2\pi}{r^2} \left( u_*(t,y) - \frac{1}{r} \right) f(t,y,r) \right) & = 0, \qquad r\in (0,\infty),
\end{align*}
for \(t>0\) and \(y\in \Omega_1\), supplemented with periodic boundary conditions on \(\Omega_1\). The parameter \(\delta\) is prescribed by the scaling, while \(f\) is in general a measure-valued solution to the above transport equation.Upscaling of a system of diffusion-reaction equations coupled with a system of ordinary differential equations originating in the context of crystal dissolution and precipitation of minerals in a porous mediumhttps://zbmath.org/1472.350292021-11-25T18:46:10.358925Z"Mahato, Hari Shankar"https://zbmath.org/authors/?q=ai:mahato.hari-shankar"Kräutle, Serge"https://zbmath.org/authors/?q=ai:krautle.serge"Knabner, Peter"https://zbmath.org/authors/?q=ai:knabner.peter"Böhm, Michael"https://zbmath.org/authors/?q=ai:bohm.michael-j|bohm.michael-cSummary: In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and precipitation of immobile species present inside a porous medium. The transport of mobile species in the pores is modeled by a system of semilinear parabolic partial differential equations. The reactions amongst the mobile species are assumed to be reversible. i.e. both forward and backward reactions are considered. These reversible reactions lead to highly nonlinear reaction rate terms on the right-hand side of the partial differential equations. This system of equations for the mobile species is complemented by flux boundary conditions at the outer boundary. Furthermore, the dissolution and precipitation of immobile species on the surface of the solid parts are modeled by mass action kinetics which lead to a nonlinear precipitation term and a multivalued dissolution term. The model is posed at the pore (micro) scale. The contribution of this paper is two-fold: first we show the existence of a unique positive global weak solution for the coupled systems and then we upscale (homogenize) the model from the micro scale to the macro scale. For the existence of solution, some regularization techniques, Schaefer's fixed point theorem and Lyapunov type arguments have been used whereas the concepts of two-scale convergence and periodic unfolding are used for the homogenization.The geometric average of curl-free fields in periodic geometrieshttps://zbmath.org/1472.350302021-11-25T18:46:10.358925Z"Poelstra, Klaas Hendrik"https://zbmath.org/authors/?q=ai:poelstra.klaas-hendrik"Schweizer, Ben"https://zbmath.org/authors/?q=ai:schweizer.ben"Urban, Maik"https://zbmath.org/authors/?q=ai:urban.maikSummary: In periodic homogenization problems, one considers a sequence \((u^{\eta})_{\eta}\) of solutions to periodic problems and derives a homogenized equation for an effective quantity \(\hat{u} \). In many applications, \( \hat{u}\) is the weak limit of \((u^{\eta})_{\eta} \), but in some applications \(\hat{u}\) must be defined differently. In the homogenization of Maxwell's equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric average has been introduced in [\textit{G. Bouchitté} et al., C. R., Math., Acad. Sci. Paris 347, No. 9--10, 571--576 (2009; Zbl 1177.35028)]; it associates to a curl-free field \(Y\setminus\overline{\Sigma}\to\mathbb{R}^3\), where \(Y\) is the periodicity cell and \(\Sigma\) an inclusion, a vector in \(\mathbb{R}^3 \). In this article, we extend previous definitions to more general inclusions, in particular inclusions that are not compactly supported in the periodicity cell. The physical relevance of the geometric average is demonstrated by various results, e.g., a continuity property of limits of tangential traces.Error estimates for elliptic equations with not-exactly periodic coefficientshttps://zbmath.org/1472.350312021-11-25T18:46:10.358925Z"Reichelt, Sina"https://zbmath.org/authors/?q=ai:reichelt.sinaSummary: This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly \(\varepsilon\)-periodic and the ellipticity constant may degenerate with order \(\mathcal{O}(\varepsilon^{2\gamma})\). Here, \(\varepsilon>0\) denotes the ratio between the microscopic and the macroscopic length scale, and the coefficients are only periodic with respect to the microscopic scale. It is shown that for \(\gamma=0\) and \(\gamma=1\) the error between the original solution and the effective solution is of order \(\mathcal{O}(\varepsilon^{1/2})\). Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale.Spectrum of one-dimensional eigenoscillations of a medium consisting of viscoelastic material with memory and incompressible viscous fluidhttps://zbmath.org/1472.350322021-11-25T18:46:10.358925Z"Shamaev, A. S."https://zbmath.org/authors/?q=ai:shamaev.alexei-s"Shumilova, V. V."https://zbmath.org/authors/?q=ai:shumilova.vladlena-valerievnaSummary: We consider the initial-boundary value problem describing joint oscillations of periodically alternating layers of viscoelastic material with memory and layers of viscous incompressible fluid. Solving auxiliary problems on the periodicity cell, we write the corresponding homogenized problem and show that the eigenvalues of the boundary value problems involved in the spectra of one-dimensional eigenoscillations of the original and homogenized media are roots of transcendental and algebraic equations respectively. In the case where the eigenoscillations are perpendicular to the layers, we show that the spectra of the original problems Hausdorff converge to the spectrum of the homogenized problem.Stochastic homogenization of a class of quasiconvex viscous Hamilton-Jacobi equations in one space dimensionhttps://zbmath.org/1472.350332021-11-25T18:46:10.358925Z"Yilmaz, Atilla"https://zbmath.org/authors/?q=ai:yilmaz.atillaSummary: We prove homogenization for a class of viscous Hamilton-Jacobi equations in the stationary \& ergodic setting in one space dimension. Our assumptions include most notably the following: the Hamiltonian is of the form \(G(p) + \beta V(x, \omega)\), the function \(G\) is coercive and strictly quasiconvex, \( \min G = 0\), \(\beta > 0\), the random potential \(V\) takes values in \([0, 1]\) with full support and it satisfies a hill condition that involves the diffusion coefficient. Our approach is based on showing that, for every direction outside of a bounded interval \(( \theta_1(\beta), \theta_2(\beta))\), there is a unique sublinear corrector with certain properties. We obtain a formula for the effective Hamiltonian and deduce that it is coercive, identically equal to \(\beta\) on \(( \theta_1(\beta), \theta_2(\beta))\), and strictly monotone elsewhere.The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion systemhttps://zbmath.org/1472.350342021-11-25T18:46:10.358925Z"Song, Yongli"https://zbmath.org/authors/?q=ai:song.yongli"Peng, Yahong"https://zbmath.org/authors/?q=ai:peng.yahong"Zhang, Tonghua"https://zbmath.org/authors/?q=ai:zhang.tonghuaSummary: The memory-based diffusion systems have wide applications in practice. Hopf bifurcations are observed from such systems. To meet the demand for computing the normal forms of the Hopf bifurcations of such systems, we develop an effective new algorithm where the memory delay is treated as the perturbation parameter. To illustrate the effectiveness of the algorithm, we consider a diffusive predator-prey system with memory-based diffusion and Holling type-II functional response. By employing this newly developed procedure, we investigate the direction and stability of the delay-induced mode-1 and mode-2 Hopf bifurcations. Numerical simulations confirm our theoretical findings, that is the existence of stable spatially inhomogeneous periodic solutions with mode-1 and mode-2 spatial patterns, and the transition from the unstable mode-2 spatially inhomogeneous periodic solution to the stable mode-1 spatially inhomogeneous periodic solution.Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interactionhttps://zbmath.org/1472.350352021-11-25T18:46:10.358925Z"Wang, Xiaoli"https://zbmath.org/authors/?q=ai:wang.xiaoli.1"Shi, Junping"https://zbmath.org/authors/?q=ai:shi.junping"Zhang, Guohong"https://zbmath.org/authors/?q=ai:zhang.guohongA reaction-diffusion model describing water and plant interaction in a flat environment is studied. The system is governed by the Klausmeier-Gray-Scott equations
\begin{align*}
\frac{\partial W}{\partial t} &= d_1 \Delta W + a - W B^2 - W, \\
\frac{\partial B}{\partial t} &= d_2 \Delta B + W B^2 - mB
\end{align*}
with Neumann boundary conditions.
After carefully investigating the existence and stability of uniform steady states the authors study the bifurcations from these steady states to large amplitude spatial patterned solutions. In the case of small rain fall \(a\) the authors construct a simpler shadow system by considering the singular limit \(d_1\rightarrow \infty\). The authors also carry out numerical investigations on linear and rectangular domains, which demonstrate the occurence of complicated vegetation patterns.
The article shows, that slow plant diffusion and fast water diffusion can support a vegetation state with vegetation concentrating in a small area or ``spots'', even when the rainfall is too low to support uniform vegetation.Turing instability of the periodic solutions for reaction-diffusion systems with cross-diffusion and the patch model with cross-diffusion-like couplinghttps://zbmath.org/1472.350362021-11-25T18:46:10.358925Z"Yi, Fengqi"https://zbmath.org/authors/?q=ai:yi.fengqiThe author studies a system of reaction-diffusion equations
\[
\frac{\partial U}{\partial t} = E \Delta U + F(U)
\]
with Neumann boundary conditions on a star-shaped domain \(\Omega_1 = \ell \Omega_0\). Also a patch model of \(n\) coupled reactors with cross-diffusion-like coupling is considered.
If the uniform steady state undergoes a Hopf bifurcation, the bifurcating periodic solutions may become unstable due to the cross-diffusion terms for sufficiently large domains.
As an example the Lengyel-Epstein model is investigated, for which the author presents explicit formulae for the Lyapunov coefficients and the Turing instability due to cross-diffusion.Stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problemhttps://zbmath.org/1472.350372021-11-25T18:46:10.358925Z"Carvalho, Alexandre N."https://zbmath.org/authors/?q=ai:nolasco-de-carvalho.alexandre"Moreira, Estefani M."https://zbmath.org/authors/?q=ai:moreira.estefani-mSummary: In this work, we present results on stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem. We show that this nonlocal version of the well-known Chafee-Infante equation bears some resemblance with the local version. However, its nonlocal characteristic requires a fine analysis of the spectrum of the associated linear operators, a lot more elaborated than the local case. The saddle point property of equilibria is shown to hold for this quasilinear model.Threshold dynamics of a vector-host epidemic model with spatial structure and nonlinear incidence ratehttps://zbmath.org/1472.350382021-11-25T18:46:10.358925Z"Duan, Lian"https://zbmath.org/authors/?q=ai:duan.lian"Huang, Lihong"https://zbmath.org/authors/?q=ai:huang.lihongSummary: In this paper, we formulate and study a vector-host epidemic model with spatial heterogeneity and general incidence rate. Our analyses show that if \(\mathcal{R}_0<1\) the disease-free steady state is globally asymptotically stable and the disease dies out; if \(\mathcal{R}_0>1\) then the disease persists. The obtained results are new and extend the previous results in the literature.Orbital stability investigations for travelling waves in a nonlinearly supported beamhttps://zbmath.org/1472.350392021-11-25T18:46:10.358925Z"Nagatou, K."https://zbmath.org/authors/?q=ai:nagatou.kaori"Plum, M."https://zbmath.org/authors/?q=ai:plum.michael"McKenna, P. J."https://zbmath.org/authors/?q=ai:mckenna.patrick-josephThis paper investigates the orbital stability of the traveling waves solutions of the following nonlinear beam equation:
\[
\varphi_{tt} + \varphi_{xxxx} + e^{\varphi} - 1 = 0, \qquad (x, t) \in \mathbb{R} \times \mathbb{R}.
\]
More precisely, this paper rigorously prove the orbital stability for one traveling wave, and orbital instability for 15 traveling waves among the 36 known traveling waves, solutions to this equation. The employed method to achieve these results combines analytical and computed-assisted techniques.Stability of elliptic solutions to the sinh-Gordon equationhttps://zbmath.org/1472.350402021-11-25T18:46:10.358925Z"Sun, Wen-Rong"https://zbmath.org/authors/?q=ai:sun.wen-rong"Deconinck, Bernard"https://zbmath.org/authors/?q=ai:deconinck.bernardSummary: Using the integrability of the sinh-Gordon equation, we demonstrate the spectral stability of its elliptic solutions. With the first three conserved quantities of the sinh-Gordon equation, we construct a Lyapunov functional. By using such Lyapunov functional, we show that these elliptic solutions are orbitally stable with respect to subharmonic perturbations of arbitrary period.Corrigendum to: ``Orbital stability for the Schrödinger operator involving inverse square potential''https://zbmath.org/1472.350412021-11-25T18:46:10.358925Z"Trachanas, Georgios P."https://zbmath.org/authors/?q=ai:trachanas.georgios-p"Zographopoulos, Nikolaos B."https://zbmath.org/authors/?q=ai:zographopoulos.nikolaos-bFrom the text: There is a gap in the proofs of [the authors, ibid. 259, No. 10, 4989--5016 (2015; Zbl 1326.35036), Theorem 3.1, p. 5000] and [loc. cit., Theorem 4.1, p. 5012] concerning uniqueness.Analysis of spatial patterns in the distributed stochastic Brusselatorhttps://zbmath.org/1472.350422021-11-25T18:46:10.358925Z"Kolinichenko, A. P."https://zbmath.org/authors/?q=ai:kolinichenko.aleksandr-pavlovich"Ryashko, L. B."https://zbmath.org/authors/?q=ai:ryashko.lev-borisovichSummary: Stochastic Brusselator model with the diffusion is studied. We show that in the zone of Turing instability a plethora of heterogeneous wave-like structures is formed. The influence of random perturbations is analyzed. We consider the scenarios of pattern formation in the zone of Turing stability as well as transitions between coexisting patterns in the instability zones.
For the entire collection see [Zbl 1467.34001].Large time behavior of solutions to Schrödinger equation with complex-valued potentialhttps://zbmath.org/1472.350432021-11-25T18:46:10.358925Z"Aafarani, Maha"https://zbmath.org/authors/?q=ai:aafarani.mahaSummary: We study the large-time behavior of the solutions to the Schrödinger equation associated with a quickly decaying potential in dimension three. We establish the resolvent expansions at threshold zero and near positive resonances. The large-time expansions of solutions are obtained under different conditions, including the existence of positive resonances and zero resonance or/and zero eigenvalue.New general decay result of the laminated beam system with infinite historyhttps://zbmath.org/1472.350442021-11-25T18:46:10.358925Z"Al-Mahdi, Adel M."https://zbmath.org/authors/?q=ai:al-mahdi.adel-m"Al-Gharabli, Mohammad M."https://zbmath.org/authors/?q=ai:algharabli.mohammad-m"Messaoudi, Salim A."https://zbmath.org/authors/?q=ai:messaoudi.salim-aSummary: This paper is concerned with the asymptotic behavior of the solution of a laminated Timoshenko beam system with viscoelastic damping. We extend the work known for this system with finite memory to the case of infinite memory. We use minimal and general conditions on the relaxation function and establish explicit energy decay formula, which gives the best decay rates expected under this level of generality. We assume that the relaxation function \(g\) satisfies, for some nonnegative functions \(\xi\) and \(H\), \(g'(t)\leq-\xi(t)H(g(t))\), \(\forall t\geq 0\). Our decay results generalize and improve many earlier results in the literature. Moreover, we remove some assumptions on the boundedness of initial data used in many earlier papers in the literature.Large time behaviour and synchronization of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo typehttps://zbmath.org/1472.350452021-11-25T18:46:10.358925Z"Ambrosio, B."https://zbmath.org/authors/?q=ai:ambrosio.benjamin"Aziz-Alaoui, M. A."https://zbmath.org/authors/?q=ai:aziz-alaoui.m-a"Phan, V. L. E."https://zbmath.org/authors/?q=ai:phan.v-l-emSummary: We focus on the long-time behaviour of complex networks of reaction-diffusion (RD) systems. In a previous work, we have proved the existence of the global attractor and the \(L^{\infty}\)-bound for these networks. Here, we discuss the synchronization phenomenon and establish the existence of a coupling strength threshold value that ensures this synchronization. Then, we apply these results to some particular networks with different structures (i.e. different topologies) and perform numerical simulations. We found out theoretical and numerical heuristic laws for the minimal coupling strength needed for synchronization with respect to the number of nodes and the network topology. We also discuss the link between spatial heterogeneity and synchronization. Our main conclusion is that some of widespread heuristic laws known for synchronization of ordinary differential equations remain valid for networks of RD systems, i.e. networks in which each node has its own spatial heterogeneity.Well-posedness and general energy decay of solution for transmission problem with weakly nonlinear dissipativehttps://zbmath.org/1472.350462021-11-25T18:46:10.358925Z"Bahri, Noureddine"https://zbmath.org/authors/?q=ai:bahri.noureddine"Abdelli, Mama"https://zbmath.org/authors/?q=ai:abdelli.mama"Beniani, Abderrahmane"https://zbmath.org/authors/?q=ai:beniani.abderrahmane"Zennir, Khaled"https://zbmath.org/authors/?q=ai:zennir.khaledSummary: In this paper, we consider a transmission problem in a bounded domain with a nonlinear dissipation in the first equation. Under suitable assumptions on the weight of the damping, we show the existence and uniqueness of solution by the Faedo-Galerkin method. Also we prove general stability estimates using some properties of convex functions and Lyaponov functional.Well-posedness and exponential stability results for a nonlinear Kuramoto-Sivashinsky equation with a boundary time-delayhttps://zbmath.org/1472.350472021-11-25T18:46:10.358925Z"Chentouf, Boumediène"https://zbmath.org/authors/?q=ai:chentouf.boumedieneSummary: The main concern of this article is to deal with the presence of a boundary delay in the nonlinear Kuramoto-Sivashinsky equation. First, we prove that the whole system is well-posed under a smallness assumption on the initial data and a condition on the parameter involved in the delay term. Then, the solution is shown to decay exponentially despite the presence of the delay. These findings are proved by considering several situations depending on the physical parameters of the system.Cauchy problem for thermoelastic plate equations with different damping mechanismshttps://zbmath.org/1472.350482021-11-25T18:46:10.358925Z"Chen, Wenhui"https://zbmath.org/authors/?q=ai:chen.wenhuiThe author studies the Cauchy problem for a thermoelastic plate equation in \(\mathbb{R}^n,\ n\ge 1\) with the heat conduction modeled by Fourier's law:
\begin{align*}
&u_{tt}+\Delta^2u + \Delta\theta + (-\Delta)^\sigma u_t=0,\quad \theta_t-\Delta \theta - \Delta u_t = 0,\ \delta \in [0,2],\quad t>0,\ x\in \mathbb{R}^n;\\
&(u,u_t,\theta)(0,x)=(u_0,u_1,\theta_0)(x),\ x\in \mathbb{R}^n
\end{align*}
with \(\sigma=0\) standing for the system with friction or external damping, \(\sigma \in (0,2]\) expresses the structural damping with Kelvin-Voigt type in the case \(\sigma = 2\). Applying the Fourier transform he achieves the smoothing effects and \(L^2\) well-posedness of the problem. Various qualitative properties of solutions involving the diffusion phenomena and their asymptotic profiles in the framework of weighted \(L^1\) data are achieved too.A heat equation with memory: large-time behaviorhttps://zbmath.org/1472.350492021-11-25T18:46:10.358925Z"Cortázar, Carmen"https://zbmath.org/authors/?q=ai:cortazar.carmen"Quirós, Fernando"https://zbmath.org/authors/?q=ai:quiros-gracian.fernando"Wolanski, Noemí"https://zbmath.org/authors/?q=ai:wolanski.noemi-iSummary: We study the large-time behavior in all \(L^p\) norms of solutions to a heat equation with a Caputo \(\alpha \)-time derivative posed in \(\mathbb{R}^N\) (\(0 < \alpha < 1\)). These are known as subdiffusion equations. The initial data are assumed to be integrable, and, when required, to be also in \(L^p\).
We find that the decay rate in all \(L^p\) norms, \(1 \leq p \leq \infty \), depends greatly on the space-time scale under consideration. This result explains in particular the so called ``critical dimension phenomenon'' (cf. [\textit{J. Kemppainen} et al., Math. Ann. 366, No. 3--4, 941--979 (2016; Zbl 1354.35178)]).
Moreover, we find the final profiles (that strongly depend on the scale). The most striking result states that in compact sets the final profile (in all \(L^p\) norms) is a multiple of the Newtonian potential of the initial datum.
Our results are very different from the ones for classical diffusion equations and show that, in accordance with the physics they have been proposed for, these are good models for particle systems with sticking and trapping phenomena or fluids with memory.Decay estimates for nonradiative solutions of the energy-critical focusing wave equationhttps://zbmath.org/1472.350502021-11-25T18:46:10.358925Z"Duyckaerts, Thomas"https://zbmath.org/authors/?q=ai:duyckaerts.thomas"Kenig, Carlos"https://zbmath.org/authors/?q=ai:kenig.carlos-e"Merle, Frank"https://zbmath.org/authors/?q=ai:merle.frankSummary: We consider the energy-critical focusing wave equation in space dimension \(N\geq 3\). The equation has a nonzero radial stationary solution \(W\), which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation, behaves asymptotically as a sum of modulated \(W\) s, decoupled by the scaling, and a radiation term. A \textit{nonradiative} solution of the equation is by definition a solution of which energy in the exterior \(\{|x|>|t|\}\) of the wave cone vanishes asymptotically as \(t\rightarrow+\infty\) and \(t\rightarrow-\infty\). In our previous work [Camb. J. Math. 1, No. 1, 75--144 (2013; Zbl 1308.35143)], we have proved that the only radial nonradiative solutions of the equation in three space dimensions are, up to scaling, 0 and \(\pm W\). This was crucial in the proof of soliton resolution in [loc. cit.]. In this paper, we prove that the initial data of a radial nonradiative solution in odd space dimension have a prescribed asymptotic behavior as \(r\rightarrow \infty\). We will use this property for the proof of soliton resolution, for radial data, in all odd space dimensions. The proof uses the characterization of nonradiative solutions of the linear wave equation in odd space dimensions obtained by Lawrie, Liu, Schlag, and the second author in [\textit{C. Kenig} et al., Adv. Math. 285, 877--936 (2015; Zbl 1331.35209)]. We also study the propagation of the support of nonzero radial solutions with compactly supported initial data and prove that these solutions cannot be nonradiative.Doubly nonlocal Fisher-KPP equation: front propagationhttps://zbmath.org/1472.350512021-11-25T18:46:10.358925Z"Finkelshtein, Dmitri"https://zbmath.org/authors/?q=ai:finkelshtein.dmitri-l"Kondratiev, Yuri"https://zbmath.org/authors/?q=ai:kondratiev.yuri-g"Tkachov, Pasha"https://zbmath.org/authors/?q=ai:tkachov.pashaThe authors study the large time propagation of solutions to a doubly nonlocal reaction-diffusion equation of the Fisher-KPP type. By doubly nonlocal it is meant that both diffusion and reaction (through competitite terms) are nonlocal. For compactly supported initial data (or more generally initial data that decay slower than some exponentials), the asymptotic shape of the propagation and the spreading speed in any direction are computed. The argument draws some inspiration from a classical scheme of Weinberger.Fundamental solutions and decay rates for evolution problems on the torus \(\mathbb{T}^n\)https://zbmath.org/1472.350522021-11-25T18:46:10.358925Z"Guiñazú, Alex"https://zbmath.org/authors/?q=ai:guinazu.alex"Vergara, Vicente"https://zbmath.org/authors/?q=ai:vergara.vicenteSummary: In this paper we study large-time behavior evolution problems on the n-dimensional torus \(\mathbb{T}^n\), \(n\geq 1\). Here we analyze the solutions to these problems, studying their regularity and obtaining estimates of them. The main tools we use is the toroidal Fourier transform, together with Fourier series and a version of the Hardy-Littlewood inequality, applied to our case of the n-dimensional torus \(\mathbb{T}^n\). We use this inequality to find an estimate of solutions to evolution problems.Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean spacehttps://zbmath.org/1472.350532021-11-25T18:46:10.358925Z"Hui, Kin Ming"https://zbmath.org/authors/?q=ai:hui.kin-ming"Park, Jinwan"https://zbmath.org/authors/?q=ai:park.jinwanSummary: For \(n\ge 3\), \(0<m<\frac{n-2}{n}\), \(\beta<0\) and \(\alpha = \frac{2\beta}{1-m}\), we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in \((\mathbb{R}^n\setminus\{0\})\times \mathbb{R}\) of the form \(U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R},\) where \(f_{\lambda}\) is a radially symmetric function satisfying
\[
\frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\},
\]
with \(\underset{r\to 0}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}} = \frac{2(n-1)(n-2-nm)}{|\beta|(1-m)}\) and \(\underset{r\to\infty}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}}\), for some constant \(\lambda>0\).
As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation \(u_t = \frac{n-1}{m}\Delta u^m\) in \((\mathbb{R}^n\setminus\{0\})\times (0,\infty)\) with initial value \(u_0\) satisfying \(f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x)\), \(\forall x\in\mathbb{R}^n\setminus\{0\}\), such that the solution \(u\) satisfies \(U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t)\), \(\forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0\), for some constants \(\lambda_1>\lambda_2>0\).
We also prove the asymptotic large time behaviour of such singular solution \(u\) when \(n = 3,4\) and \(\frac{n-2}{n+2}\le m<\frac{n-2}{n}\) holds. Asymptotic large time behaviour of such singular solution \(u\) is also obtained when \(3\le n<8\), \(1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right)\), and \(u(x,t)\) is radially symmetric in \(x\in\mathbb{R}^n\setminus\{0\}\) for any \(t>0\) under appropriate conditions on the initial value \(u_0\).Chemotaxis with quadratic dissipation and logistic sourcehttps://zbmath.org/1472.350542021-11-25T18:46:10.358925Z"Latos, Evangelos"https://zbmath.org/authors/?q=ai:latos.evangelos-a"Suzuki, Takashi"https://zbmath.org/authors/?q=ai:suzuki.takashiSummary: We study systems of chemotaxis with quadratic dissipation and logistic source. Global existence and convergence to the spatially homogeneous steady state are proven in several cases, particularly when the space dimension is two or the dissipation dominates the chemotaxis. The blowup case is also studied for modified nonlinearities.Asymptotics for 2-D wave equations with Wentzell boundary conditions in the squarehttps://zbmath.org/1472.350552021-11-25T18:46:10.358925Z"Li, Chan"https://zbmath.org/authors/?q=ai:li.chan"Jin, Kun-Peng"https://zbmath.org/authors/?q=ai:jin.kunpengThis paper deals with the linear wave equations with frictional dampings on Wentzell boundary,
\[
\left\{\!\!\! \begin{array}{lll} &u_{tt}-\Delta u=0, &x\in\Omega,\> t\ge0,\\
&u=0, &x\in\Gamma_0,\> t\ge0,\\
&u_{tt}-\Delta_Tu+\partial_\nu u+u_t=0, &x\in\Gamma_1,\> t\ge0,\\
&u(0,x) = u_0(x),\>u_t(0,x) = u_1(,x),&x\in \Omega, \end{array} \right.
\]
which models the vertical motion of membranes edged with a thin boundary coil of high rigidity. Here,
\[
\begin{array}{ll} &\Omega=(0,1)\times(0,1),\\
&\Gamma_0 = \{(x,1):\, 0 < x < 1\}\cup\{(0,y):\, 0 < y < 1\},\\
&\Gamma_1 = \{(x,0):\, 0 < x < 1\} \cup \{(1, y):\, 0 < y < 1\}, \end{array}
\]
\(\nu(\cdot)\) is the unit outer normal vector at boundary, \(\Delta_T\) is the tangential Laplacian operator on \(\Gamma_1\). The motion at the boundary \(\Gamma_1\) is constrained to vertical motion and is governed by the Newton equation of motion. The boundary condition on \(\Gamma_1\) is a dynamic Wentzell boundary one. The authors study the stabilization of the problem.Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensionshttps://zbmath.org/1472.350562021-11-25T18:46:10.358925Z"Masaki, Satoshi"https://zbmath.org/authors/?q=ai:masaki.satoshi"Miyazaki, Hayato"https://zbmath.org/authors/?q=ai:miyazaki.hayato"Uriya, Kota"https://zbmath.org/authors/?q=ai:uriya.kotaSummary: In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [SIAM J. Math. Anal. 50, No. 3, 3251--3270 (2018; Zbl 1397.35284)], the first and second authors consider one- and two-dimensional cases and give a sufficient condition on the nonlinearity so that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converge to a given asymptotic profile in a faster rate than in the lower dimensional cases. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in the aforementioned article. Moreover, we present a candidate for the second asymptotic profile of the solution.Global wellposedness and large time behavior of solutions to the \(N\)-dimensional compressible Oldroyd-B modelhttps://zbmath.org/1472.350572021-11-25T18:46:10.358925Z"Zhai, Xiaoping"https://zbmath.org/authors/?q=ai:zhai.xiaoping"Li, Yongsheng"https://zbmath.org/authors/?q=ai:li.yongshengThe authors consider the compressible Oldroyd-B model written as \(\partial _{t}a+\operatorname{div} u=-\operatorname{div}(au)\), \(\partial _{t}\eta -\varepsilon \Delta \eta =-\operatorname{div}(\eta u)\), \(\partial _{t}\mathbb{T}+\frac{A_{0}}{2\lambda _{1}}\mathbb{T}+(u\cdot \nabla )\mathbb{T}-\varepsilon \Delta \mathbb{T}=\frac{\kappa A_{0}}{ 2\lambda _{1}}\eta \mathbb{I}d+F(\mathbb{T},u)\), \(\partial _{t}u+u\cdot \nabla u-\mu \Delta u-(\lambda +\mu )\nabla \operatorname{div}u+\nabla a=\operatorname{div}\mathbb{T} -\kappa L\nabla \eta +G(a,u,\eta ,\mathbb{T})\), with \(F(\mathbb{T} ,u)=(\nabla u\mathbb{T}+\mathbb{T}\nabla u)-\mathbb{T}\operatorname{div}u\) and \(G(a,u,\eta , \mathbb{T})=k(a)\nabla a-I(a)(\mu u+(\lambda +\mu )\nabla \operatorname{div}u)-I(a)(\operatorname{div}\mathbb{T}-\nabla \eta )-\zeta (1-I(a))\eta \nabla \eta \); posed in \(\mathbb{R}^{+}\times \mathbb{R}^{n}\), \(n=2,3\). Here \(a=1-\rho \), \(\rho = \rho (t,x)\in \mathbb{R}^{+}\) is the density function of the fluid, \(u=u(t,x)\in \mathbb{R}^{n}\) is the velocity. \(\mathbb{T}=(\mathbb{T}_{i,j}(t,x))\), \( 1\leq i,j\leq n\), is a symmetric matrix function representing the extra stress tensor, \(\eta =\eta (t,x)\in \mathbb{R}^{+}\) represents the polymer number density defined as the integral of a probability density function \( \psi \) which is governed by the Fokker-Plank equation, with respect to the conformation vector, \(\mu >0\) and \(\lambda \) are the viscosity constants which are supposed to satisfy \(n\lambda +2\mu \geq 0\), the parameters \( \kappa \), \(\varepsilon \), \(A_{0}\), \(\lambda _{1}\) are positive numbers, whereas \(\zeta \geq 0\) and \(L\geq 0\) with \(\zeta +L\neq 0\). Initial conditions are added to \((a,u,\eta ,\mathbb{T})\). The authors first prove a local existence result for this problem, assuming \(1<p<2n\) and that for any initial data \(u_{0}\in \overset{.}{B}_{p,1}^{n/p-1}(\mathbb{R}^{n})\), \((\eta _{0},\mathbb{T}_{0})\in \overset{.}{B}_{p,1}^{n/p}(\mathbb{R}^{n})\) and \( a_{0}\in \overset{.}{B}_{p,1}^{n/p}(\mathbb{R}^{n})\) with \(1+a_{0}\) bounded away from zero.\ There exists \(T>0\) such that the above problem has a unique solution with
\begin{align*}
& a\in C_{b}([0,T];\overset{.}{B}_{p,1}^{n/p}),\; u\in C_{b}([0,T];\overset{.}{B}_{p,1}^{n/p-1})\cap L^{1}([0,T];\overset{.}{B} _{p,1}^{n/p+1}), \\
& (\eta ,\mathbb{T})\in C_{b}([0,T];\overset{.}{B} _{p,1}^{n/p})\cap L^{1}([0,T];\overset{.}{B}_{p,1}^{n/p+2}), \; \mathbb{T}\in L^{1}([0,T];\overset{.}{B}_{p,1}^{n/p}).
\end{align*}
The authors then prove a global existence result to this problem, assuming further hypotheses on \(p\) and on the initial data. The last main result of the problem proves optimal decay estimates on this global solution with respect to time. The authors observe that the equations of the polymer number density \(\eta \) and the extra stress tensor \(\mathbb{T}\) are two heat-flow type, and that, when the polymer number density \(\eta \) and the extra stress tensor \(\mathbb{T}\) vanish, the above problem reduces to the barotropic Navier-Stokes equations. They thus refer to previous results of the literature for the local existence result. For the proof of the second main result, the authors use a continuity argument, establishing a priori bounds considering the low frequency and the high frequency parts of the solution. For the decay estimates, the authors establish a Lyapunov-type inequality in time for energy norms.Global classical solutions and stabilization in a two-dimensional parabolic-elliptic Keller-Segel-Stokes systemhttps://zbmath.org/1472.350582021-11-25T18:46:10.358925Z"Zheng, Jiashan"https://zbmath.org/authors/?q=ai:zheng.jiashanSummary: A class of parabolic-elliptic Keller-Segel-Stokes systems generalizing the prototype
\[
\begin{cases}
n_t+u\cdot\nabla n=\Delta n-C_S\nabla\cdot(n(1+n)^{-\alpha}\nabla c),\quad x\in\Omega,t>0,\\
u\cdot\nabla c=\Delta c-c+n,\quad x\in\Omega,t>0,\\
u_t+\nabla P=\Delta u+n\nabla\phi,\quad x\in\Omega,t>0,\\
\nabla\cdot u=0,\quad x\in\Omega,t>0,
\end{cases}\tag{\(KSF\)}
\]
is considered under boundary conditions of homogeneous Neumann type for \(n\) (the density of the cell population) and \(c\) (the chemical concentration), and Dirichlet type for \(u\) (the velocity field), in a bounded domain \(\Omega \subseteq\mathbb{R}^2\) with smooth boundary, where \(C_S>0\) and \(\phi\) is a given sufficiently smooth function. The model is proposed to describe chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells. Moreover, the chemical diffuses much faster than the cells move. It is shown that under the condition that
\[
\alpha >0,
\]
for any sufficiently smooth initial data \((n_0,u_0)\) satisfying some compatibility conditions, the associated initial-boundary-value problem \((KSF)\) possesses a global bounded classical solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on \(\alpha\) here is optimal. Building on this boundedness property, it can finally even be proved that the corresponding solution of the system decays to \((\bar{n}_0,\bar{n}_0,0)\) exponentially if \(C_S\) is smaller, where \(\bar{n}_0=\frac{1}{|\Omega |}\int_{\Omega}n_0\). Our main tool is consideration of the energy functional
\[
\int_{\Omega}n^{1+\alpha},
\]
which is a new energy-like functional.On the two dimensional fast phase transition equation: well-posedness and long-time dynamicshttps://zbmath.org/1472.350592021-11-25T18:46:10.358925Z"Khanmamedov, Azer"https://zbmath.org/authors/?q=ai:khanmamedov.azer-khSummary: We consider the initial boundary value problem for the two dimensional equation describing the evolution of the systems with the fast phase transition. Under critical growth and dissipativity conditions on the nonlinearities, we prove the existence of the global attractor. In particular, we establish the existence of the global attractor for the strong solutions of the weakly damped 2D Kirchhoff plate equation involving \(p\)-Laplacian operator with the critical exponent \(p=5\), and thereby improve the previously obtained results. We also show that the approach of this paper can be applied to the hyperbolic relaxation of the 2D Cahn-Hilliard equation with the critical quartic nonlinearity.Unbounded mass radial solutions for the Keller-Segel equation in the diskhttps://zbmath.org/1472.350602021-11-25T18:46:10.358925Z"Bonheure, Denis"https://zbmath.org/authors/?q=ai:bonheure.denis"Casteras, Jean-Baptiste"https://zbmath.org/authors/?q=ai:casteras.jean-baptiste"Román, Carlos"https://zbmath.org/authors/?q=ai:roman.carlosSummary: We consider the boundary value problem
\[
\begin{cases}
-\Delta u+ u -\lambda e^u=0,\;u>0 &\text{in } B_1(0)\\
\qquad\qquad\quad\,\partial_\nu u=0 & \text{on } \partial B_1(0),
\end{cases}
\]
whose solutions correspond to steady states of the Keller-Segel system for chemotaxis. Here \(B_1(0)\) is the unit disk, \(\nu\) the outer normal to \(\partial B_1(0)\), and \(\lambda>0\) is a parameter. We show that, provided \(\lambda\) is sufficiently small, there exists a family of radial solutions \(u_\lambda\) to this system which blow up at the origin and concentrate on \(\partial B_1(0)\), as \(\lambda \rightarrow 0\). These solutions satisfy
\[
\lim_{\lambda\rightarrow 0} \frac{u_\lambda (0)}{|\ln \lambda|}=0\quad \text{and}\quad 0<\lim_{\lambda\rightarrow 0} \frac{1}{|\ln \lambda |}\int_{B_1(0)}\lambda e^{u_\lambda (x)}dx<\infty,
\]
having in particular unbounded mass, as \(\lambda \rightarrow 0\).Blow-up phenomena in a parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with superlinear logistic degradationhttps://zbmath.org/1472.350612021-11-25T18:46:10.358925Z"Chiyo, Yutaro"https://zbmath.org/authors/?q=ai:chiyo.yutaro"Marras, Monica"https://zbmath.org/authors/?q=ai:marras.monica"Tanaka, Yuya"https://zbmath.org/authors/?q=ai:tanaka.yuya"Yokota, Tomomi"https://zbmath.org/authors/?q=ai:yokota.tomomiSummary: This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation,
\[
\begin{cases}
u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot (u\nabla w)+\lambda u-\mu u^k,\quad & x\in\Omega,\, t>0, \\
0=\Delta v+\alpha u-\beta v,\quad & x\in\Omega,\, t>0, \\
0=\Delta w+\gamma u-\delta w,\quad & x\in\Omega,\, t>0,
\end{cases}
\]
under homogeneous Neumann boundary conditions, in a ball \(\Omega\subset\mathbb{R}^n\) \((n\geq 3)\), with constant parameters \(\lambda\in\mathbb{R}\), \(k>1\), \(\mu,\chi,\xi,\alpha,\beta,\gamma,\delta>0\). Blow-up phenomena in the system have been well investigated in the case \(\lambda=\mu=0\), whereas the attraction-repulsion chemotaxis system with logistic degradation has been not studied. Under the condition that \(k>1\) is close to 1, this paper ensures a solution which blows up in \(L^\infty\)-norm and \(L^\sigma\)-norm with some \(\sigma>1\) for some nonnegative initial data. Moreover, a lower bound of blow-up time is derived.New blow-up criterion for the Degasperis-Procesi equation with weak dissipationhttps://zbmath.org/1472.350622021-11-25T18:46:10.358925Z"Deng, Xijun"https://zbmath.org/authors/?q=ai:deng.xijunSummary: In this paper, we investigate the Cauchy problem of the Degasperis-Procesi equation with weak dissipation. We establish a new local-in-space blow-up criterion of the dissipative Degasperis-Procesi equation on line \(\mathbb{R}\) and on circle \(S\), respectively.Correction to: ``Scattering threshold for the focusing nonlinear Klein-Gordon equation''https://zbmath.org/1472.350632021-11-25T18:46:10.358925Z"Ibrahim, Slim"https://zbmath.org/authors/?q=ai:ibrahim.slim"Masmoudi, Nader"https://zbmath.org/authors/?q=ai:masmoudi.nader"Nakanishi, Kenji"https://zbmath.org/authors/?q=ai:nakanishi.kenji.1Summary: This article resolves some errors in the paper [the authors, ibid. 4, No. 3, 405--460 (2011; Zbl 1270.35132)]. The errors are in the energy-critical cases in two and higher dimensions.Life-span of blowup solutions to semilinear wave equation with space-dependent critical dampinghttps://zbmath.org/1472.350642021-11-25T18:46:10.358925Z"Ikeda, Masahiro"https://zbmath.org/authors/?q=ai:ikeda.masahiro"Sobajima, Motohiro"https://zbmath.org/authors/?q=ai:sobajima.motohiroSummary: This paper is concerned with the blowup phenomena for the initial value problem of the wave equation with a critical space-dependent damping term \(V_0|x|^{-1}\) and a \(p\)-th order power nonlinearity on the Euclidean space \(\boldsymbol{R}^N\), where \(N \geq 3\) and \(V_0 \in [0,(N-1)^2/(N+1))\). The main result of the present paper is to prove existence of a unique local solution of the problem and to provide a sharp estimate for lifespan for such a solution for small data with a compact support when \(N/(N-1) < p \leq p_S(N + V_0)\), where \(p_S(N)\) is the Strauss exponent for the semilinear wave equation without damping. The main idea of the proof is due to the technique of test functions for the classical wave equation originated by \textit{Y. Zhou} and \textit{W. Han} [Commun. Partial Differ. Equations 39, No. 3, 439--451 (2014; Zbl 1297.35141)]. Consequently, the result poses whether the value \(V_0 = (N-1)^2/(N + 1)\) is a threshold for diffusive structure of the singular damping |\(x|^{-1}\) or not.Blow-up phenomenon for a reaction-diffusion equation with weighted nonlocal gradient absorption termshttps://zbmath.org/1472.350652021-11-25T18:46:10.358925Z"Liang, Mengyang"https://zbmath.org/authors/?q=ai:liang.mengyang"Fang, Zhong Bo"https://zbmath.org/authors/?q=ai:fang.zhongboSummary: This paper deals with the blow-up phenomenon of solutions to a reaction-diffusion equation with weighted nonlocal gradient absorption terms in a bounded domain. Based on the method of auxiliary function and the technique of modified differential inequality, we establish appropriate conditions on weight function and nonlinearities to guarantee the solution exists globally or blows up at finite time. Moreover, upper and lower bounds for blow-up time are derived under appropriate measure in higher dimensional spaces.The local behavior of positive solutions for higher order equation with isolated singularitieshttps://zbmath.org/1472.350662021-11-25T18:46:10.358925Z"Li, Yimei"https://zbmath.org/authors/?q=ai:li.yimeiSummary: We use blow up analysis for local integral equations to provide a blow up rates of solutions of higher order Hardy-Hénon equation in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This work generalizes the correspondence results of \textit{T. Jin} and \textit{J. Xiong} (in [``Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities'', Preprint, \url{arXiv:1901.01678}]) on higher order conformally invariant equations with an isolated singularity.Semilinear wave equation on compact Lie groupshttps://zbmath.org/1472.350672021-11-25T18:46:10.358925Z"Palmieri, Alessandro"https://zbmath.org/authors/?q=ai:palmieri.alessandroSummary: In this note, we study the semilinear wave equation with power nonlinearity \(|u|^p\) on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a blow-up result for the semilinear Cauchy problem for any \(p>1\), under suitable sign assumptions for the initial data. Furthermore, sharp lifespan estimates for local (in time) solutions are derived.Blow-up for a semilinear heat equation with Fujita's critical exponent on locally finite graphshttps://zbmath.org/1472.350682021-11-25T18:46:10.358925Z"Wu, Yiting"https://zbmath.org/authors/?q=ai:wu.yitingSummary: Let \(G=(V,E)\) be a locally finite, connected and weighted graph. We prove that, for a graph satisfying curvature dimension condition \(CDE'(n,0)\) and uniform polynomial volume growth of degree \(m\), all non-negative solutions of the equation \(\partial_tu=\Delta u+u^{1+\alpha}\) blow up in a finite time, provided that \(\alpha =\frac{2}{m} \). We also consider the blow-up problem under certain conditions for volume growth and initial value. These results complement our previous work joined with Lin.Blow-up of solutions to a fourth-order parabolic equation with/without \(p\)-Laplacian and general nonlinearity modeling epitaxial growthhttps://zbmath.org/1472.350692021-11-25T18:46:10.358925Z"Zhou, Jun"https://zbmath.org/authors/?q=ai:zhou.jun.1|zhou.jun.2|zhou.jun.3Summary: This paper deals with a fourth-order parabolic equation with/without \(p\)-Laplacian and general nonlinearity modeling epitaxial growth. By using the variational structure of the problem and differential inequalities, it is shown, under some conditions on the initial value, the solutions to the problem will blow up in finite time. Furthermore, the upper bound of the blow-up time for blowing-up solution is given. Moreover, the existence of a ground-state solution is obtained under approximate assumptions on the nonlinear term. The results of this paper extend and generalize some results got in the papers [\textit{G. A. Philippin}, Proc. Am. Math. Soc. 143, No. 6, 2507--2513 (2015; Zbl 1316.35151)], [\textit{Y. Han}, Nonlinear Anal., Real World Appl. 43, 451--466 (2018; Zbl 06892735)], and [the author, ibid. 48, 54--70 (2019; Zbl 1460.74061)].Modular maximal estimates of Schrödinger equationshttps://zbmath.org/1472.350702021-11-25T18:46:10.358925Z"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punSummary: This paper offers the maximal estimates of the solutions of some initial value problems on modular spaces. Our results include the estimates for the solutions of Schrödinger equation.Growth theorems for metric spaces with applications to PDEhttps://zbmath.org/1472.350712021-11-25T18:46:10.358925Z"Safonov, M. V."https://zbmath.org/authors/?q=ai:safonov.mikhail-vIn this article, the author focuses mainly on the extent of the results and techniques of him, obtained with N. V. Krylov, for the second order elliptic and parabolic equations in the form of non-divergence, which date back to 1978--1980 to a more general abstract setting in terms of metric spaces. The author deals only with the elliptic versions of these results, the extension of the results to the parabolic (time-dependent) case requires more assumptions, which are not considered here. Some elements of the techniques are similar to those previously introduced by De Giorgi, Nash and Moser for the elliptic and parabolic equations in the form of divergence. The main tools are special Landis-type growth theorems.Maximum principles and ABP estimates to nonlocal Lane-Emden systems and some consequenceshttps://zbmath.org/1472.350722021-11-25T18:46:10.358925Z"Leite, Edir Junior Ferreira"https://zbmath.org/authors/?q=ai:leite.edir-ferreira-junSummary: This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane-Emden system involving fractional Laplace operators:
\[\left\{
\begin{aligned}
(-\Delta)^s u &= \lambda\rho(x) |v|^{\alpha-1}v \\
(-\Delta)^t v &= \mu\tau(x) |u|^{\beta-1}u \\
u &= v = 0
\end{aligned} \quad
\begin{aligned}
&\text{in } \Omega, \\
&\text{in } \Omega, \\
&\text{in }\mathbb{R}^n \setminus \Omega,
\end{aligned}
\right.\]
where \(s,t\in(0,1), \alpha,\beta>0\) satisfy \(\alpha\beta=1, \Omega\) is a smooth bounded domain in \(\mathbb{R}^n, n\geq 1\), and \(\rho\) and \(\tau\) are continuous functions on \(\overline{\Omega}\) and positive in \(\Omega \). We establish some maximum principles depending on \(\Omega \). In particular, we explicitly characterize the measure of \(\Omega\) for which the maximum principles corresponding to this problem hold in \(\Omega \). For this, we derived an explicit lower estimate of principal eigenvalues in terms of the measure of \(\Omega \). Aleksandrov-Bakelman-Pucci (ABP) type estimates for the above systems are also proved. We also show the existence of a viscosity solution for a nonlinear perturbation of the nonhomogeneous counterpart of the above problem with polynomial and exponential growths. As an application of the maximum principles, we measure explicitly how small \(|\Omega|\) has to be to ensure the positivity of the obtained solutions.On the strong maximum principle for a fractional Laplacianhttps://zbmath.org/1472.350732021-11-25T18:46:10.358925Z"Trong, Nguyen Ngoc"https://zbmath.org/authors/?q=ai:trong.nguyen-ngoc"Tan, Do Duc"https://zbmath.org/authors/?q=ai:tan.do-duc"Thanh, Bui Le Trong"https://zbmath.org/authors/?q=ai:thanh.bui-le-trongLet \(\Omega\) be a bounded open connected set in \(\mathbb{R}^n\) with Lipschitz boundary, \(s\in (\frac{1}{2},1)\), \(\{e_k\}\) be the sequence of eigenfunctions of the Laplace operator \(-\Delta: H_0^2(\Omega)\rightarrow L^2(\Omega)\), and \(\{\lambda_k\}\) be the sequence of corresponding eigenvalues.
The authors establish a strong maximum principle for the spectral Dirichlet Laplacian \((-\Delta)^s\), defined by \[(-\Delta)^su=\sum_{k\in \mathbb{N}}\lambda_{k}^s\langle e_k,u\rangle_{L^2(\Omega)}e_k\] for each \(u=L^2(\Omega)\) such that \(\sum_{k\in \mathbb{N}}u_k^2\lambda_k^2<\infty\), where \(u_k\) is the component of \(u\) along \(e_k\).
In particular, the authors prove that every nonnegative function \(u\in L^1(\Omega)\) such that \((-\Delta)^su\) is a Radon measure on \(\Omega\) is almost everywhere equal to a quasi continuous function \(\tilde{u}\), and if \(\tilde{u}=0\) on a subset of \(\Omega\) with positive \(H^s\)-capacity and \(u\) satisfies \[(-\Delta)^su+au\geq 0 \ \ \ \text{a.e. in} \ \ \Omega,\] for some nonnegative function \(a\in L^1(\Omega)\), then \(u=0\) in \(\Omega\).
Ingredients of the proof are a fractional version of the Poincaré's inequality and truncated functions.Unique solvability of the stationary complex heat transfer problem in a system of grey bodies with semitransparent inclusionshttps://zbmath.org/1472.350742021-11-25T18:46:10.358925Z"Amosov, A. A."https://zbmath.org/authors/?q=ai:amosov.andreiSummary: We establish the unique solvability of a radiative-conductive heat transfer problem in a system of grey bodies with semitransparent inclusions and prove the comparison theorem. We show that higher summability of the data improves the properties (exponential summability and boundedness) of the solution.Local regularity for the harmonic map and Yang-Mills heat flowshttps://zbmath.org/1472.350752021-11-25T18:46:10.358925Z"Afuni, Ahmad"https://zbmath.org/authors/?q=ai:afuni.ahmadSummary: We establish new local regularity results for the harmonic map and Yang-Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by \textit{K. Ecker} [Calc. Var. Partial Differ. Equ. 23, No. 1, 67--81 (2005; Zbl 1119.35026)] and the author [ibid. 55, No. 1, Paper No. 13, 14 p. (2016; Zbl 1338.35020); Adv. Calc. Var. 12, No. 2, 135--156 (2019; Zbl 1415.58016)].Corrigendum to: ``Investigation of the analyticity of dissipative-dispersive systems via a semigroup method''https://zbmath.org/1472.350762021-11-25T18:46:10.358925Z"Ioakim, Xenakis"https://zbmath.org/authors/?q=ai:ioakim.xenakis"Smyrlis, Yiorgos-Sokratis"https://zbmath.org/authors/?q=ai:smyrlis.yiorgos-sokratisCorrects [the authors, ibid. 420, No. 2, 1116--1128 (2014; Zbl 1304.35172)].Remarks on parabolic De Giorgi classeshttps://zbmath.org/1472.350772021-11-25T18:46:10.358925Z"Liao, Naian"https://zbmath.org/authors/?q=ai:liao.naianIn ths very interesting paper, the author makes several remarks concerning properties of functions in parabolic De Giorgi classes of order p. This effort is motivated by the fact that There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, the author is able to give new proofs of known properties. In particular, local boundedness and local Hölder continuity of these functions via Moser's ideas, thus avoiding De Giorgi's heavy machinery, are proved. The author also takes the opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, surprisingly without any covering argument.Hölder regularity for porous medium systemshttps://zbmath.org/1472.350782021-11-25T18:46:10.358925Z"Liao, Naian"https://zbmath.org/authors/?q=ai:liao.naianIn this paper, the author considers locally bounded weak solutions to certain parabolic systems of porous medium type, i.e.
\[
\partial_t u-\mathrm{div}(m|u|^{m-1}Du) = 0,
\]
with \( m > 0\). As it is well known, everywhere regularity is in general not expected from systems of elliptic or parabolic type, nevertheless, the special quasi-diagonal structure of thus system, enables the author to achieve the Hölder regularity of locally bounded solutions. As a consequence of the local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order termshttps://zbmath.org/1472.350792021-11-25T18:46:10.358925Z"Magliocca, Martina"https://zbmath.org/authors/?q=ai:magliocca.martinaSummary: We want to analyze both regularizing effect and long, short time decay concerning a class of parabolic equations having first order superlinear terms. The model problem is the following:
\[
\begin{cases} u_t-\operatorname{div}(A(t,x)|\nabla u|^{p-2}\nabla u)=\gamma |\nabla u|^q & \text{in } (0,T)\times \Omega ,\\
u=0 &\text{on } (0,T)\times \partial \Omega ,\\
u(0,x)=u_0(x) &\text{in } \Omega ,
\end{cases}
\]
where \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\), \(N\ge 2\), \(0<T\le \infty\), \(1<p<N\) and \(q<p\). We assume that \(A(t, x)\) is a coercive, bounded and measurable matrix, the growth rate \(q\) of the gradient term is superlinear but still subnatural, \(\gamma\) is a positive constant, and the initial datum \(u_0\) is an unbounded function belonging to a well precise Lebesgue space \(L^\sigma (\Omega )\) for \(\sigma =\sigma (q,p,N)\).Partial regularity for parabolic systems with VMO-coefficientshttps://zbmath.org/1472.350802021-11-25T18:46:10.358925Z"Mons, Leon"https://zbmath.org/authors/?q=ai:mons.leonThe author studies the regularity properties of the solutions of the following parabolic system in divergence form
\[
u_t-\mathrm{div} A(x,t,u,Du)=0 \qquad \text{ in } \Omega_T=\Omega\times(-T,0))
\]
It is established a partial Hölder regularity result for the weak solutions imposing standard \(p\)-growth condition and ellipticity condition on the principal operator. The vector-function \(A\) is discontinuous in \((x,t)\) verifying a VMO-type condition.
The parabolic structure of the operator requires the use of suitable intrinsic cylinders that permit to apply the method of \(\mathcal{A}\)-caloric approximation.Quantitative regularity for \(p\)-minimizing maps through a Reifenberg theoremhttps://zbmath.org/1472.350812021-11-25T18:46:10.358925Z"Vedovato, Mattia"https://zbmath.org/authors/?q=ai:vedovato.mattiaSummary: In this article we extend to arbitrary \(p\)-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case \(p=2\). We first show that the set of singular points of such a map can be \textit{quantitatively stratified}: we classify singular points based on the number of \textit{almost-symmetries} of the map around them, as done in [\textit{J. Cheeger} and \textit{A. Naber}, Commun. Pure Appl. Math. 66, No. 6, 965--990 (2013; Zbl 1269.53063)]. Then, adapting the work of \textit{A. Naber} and \textit{D. Valtorta} [Ann. Math. (2) 185, No. 1, 131--227 (2017; Zbl 1393.58009)], we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is \(k\)-rectifiable for a \(k\) which only depends on \(p\) and the dimension of the domain.Gevrey regularity for the Vlasov-Poisson systemhttps://zbmath.org/1472.350822021-11-25T18:46:10.358925Z"Velozo Ruiz, Renato"https://zbmath.org/authors/?q=ai:velozo-ruiz.renatoIn this paper the Vlasov-Poisson system is studied in the context of the Gevrey regularity. Such type of regularity is used to describe the situation when the solutions are \(C^\infty\) but fail to be analytic. This paper offers several results in that direction.
Assuming the existence of a unique Sobolev solution for the Vlasov-Poisson system on \( \mathbb{T}^d \times \mathbb{R}^d \), the propagation of Gevrey regularity is given through appropriate quantitative estimates of the corresponding Gevrey norm and the decay of related regularity radius. Among other things, the author uses Landau damping properties for Gevrey data, see [\textit{J. Bedrossian} et al., Ann. PDE 2, No. 1, Paper No. 4, 71 p. (2016; Zbl 1402.35058)].
As an application, global existence of analytic and Gevrey solutions for the Vlasov-Poisson system on \( \mathbb{T}^3 \times \mathbb{R}^3 \) is proved. This answers a question from [\textit{S. Benachour}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 16, No. 1, 83--104 (1989; Zbl 0702.35042)].
The introduction section gives a very nice historical survey together with relevant references, putting the original results in a broader contexts.Solution of mixed half-plane diffraction problems of non-stationary plane and cylindrical waves and blast wave protection by barriershttps://zbmath.org/1472.350832021-11-25T18:46:10.358925Z"Israilov, M. Sh."https://zbmath.org/authors/?q=ai:israilov.m-sh"Nosov, S. Ye."https://zbmath.org/authors/?q=ai:nosov.s-ye"Khadisov, M.-R. B."https://zbmath.org/authors/?q=ai:khadisov.m-r-bSummary: Exact analytical solutions of diffraction problems are obtained for nonstationary plane and cylindrical waves on a half-plane at following mixed boundary conditions: the Neumann condition on one side of the half-plane and the Dirichlet condition on the other one. The numerical analysis of near-front asymptotic solutions shows that there is a considerably greater attenuation of blast waves behind the half-plane (barrier) with sides of different noise absorption properties, than in the case of complete reflection from both sides of the half-plane.Stability and uniqueness of self-similar profiles in \(L^1\) spaces for perturbations of the constant kernel in Smoluchowski's coagulation equationhttps://zbmath.org/1472.350842021-11-25T18:46:10.358925Z"Throm, Sebastian"https://zbmath.org/authors/?q=ai:throm.sebastianWhen the coagulation kernel \(K\) is homogeneous with a degree strictly smaller than one, it is expected that solutions to the coagulation equation
\begin{align*}
\partial_t \phi(t,\xi) & = \frac{1}{2} \int_0^\xi K(\xi-\eta,\eta) \phi(t,\xi-\eta) \phi(t,\eta)\ d\eta \\
& \qquad - \int_0^\infty K(\xi,\eta) \phi(t,\xi) \phi(t,\eta)\ d\eta
\end{align*}
where \((t,\xi)\in (0,\infty)\times (0,\infty)\), with non-negative initial condition \(\phi_0\in L^1((0,\infty),\xi d\xi)\), behave in a self-similar way as \(t\to\infty\). This conjecture is up to now known to be true for the constant kernel \(K(\xi,\eta)=2\), see [\textit{G. Menon} and \textit{R. L. Pego}, Commun. Pure Appl. Math. 57, No. 9, 1197--1232 (2004; Zbl 1049.35048)]. An important step in the proof of this conjecture is the uniqueness of finite mass self-similar solutions, and the latter is shown in the paper under review for small perturbations of the constant kernel with homogeneity zero, thereby improving previous results by \textit{B. Niethammer}, the author and \textit{J. J. L. Velázquez} [``A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant'', Preprint, \url{arXiv:1510.03361}; J. Stat. Phys. 164, No. 2, 399--409 (2016; Zbl 1356.82034)], along with providing a new approach. More precisely, let \(W\in C((0,\infty)^2)\) be a symmetric function satisfying
\[
0 \le W(x,y) \le \left( \frac{x}{y} \right)^\alpha + \left( \frac{y}{x} \right)^\alpha \text{ and } W(\lambda x,\lambda y) = W(x,y), \qquad (\lambda,\xi,\eta)\in (0,\infty)^3,
\]
for some \(\alpha\in (0,1)\), and set \(K_\varepsilon = 2 + \varepsilon W\) for \(\varepsilon\ge 0\). Given \(a>-1\), \(b>0\), and \(\delta>0\), it is first shown that, for \(\varepsilon>0\) sufficiently small, any self-similar profile \(u^{(\varepsilon)}\) with unit total mass satisfies
\[
\int_0^\infty \left| u^{(\varepsilon)}(x) - e^{-x} \right| (x^a + x^b)\ dx \le \delta.
\]
We recall that \(u^{(\varepsilon)}\) is a self-similar profile if \((t,\xi) \mapsto (1+t)^{-2} u^{(\varepsilon)}(\xi(1+t)^{-1})\) is a solution to the coagulation equation with kernel \(K_\varepsilon\) and that \((t,\xi)\mapsto (1+t)^{-2} e^{-\xi(1+t)^{-1}}\) is the unique self-similar solution with unit total mass to the coagulation equation with constant kernel \(K_0\). Assuming further that
\[
W(x,y)\ge c_* \left( \frac{x}{y} \right)^\alpha + \left( \frac{y}{x} \right)^\alpha, \qquad (x,y)\in (0,\infty)^2,
\]
for some \(c_*>0\) when \(\alpha\in [1/2,1)\), uniqueness of the self-similar profile with unit total mass is also established, still for \(\varepsilon\) small enough.
We also refer to [\textit{J. A. Cañizo} and the author, J. Differ. Equations 270, 285--342 (2021; Zbl 1456.35065)] for attracting properties of these self-similar solutions under similar assumptions.Convergence to traveling waves for time-periodic bistable reaction-diffusion equationshttps://zbmath.org/1472.350852021-11-25T18:46:10.358925Z"Ding, Weiwei"https://zbmath.org/authors/?q=ai:ding.weiweiThe author considers a nonautonomous reaction-diffusion equation of the time-periodic and bistable type. He proposes a new proof of convergence toward the traveling waves. This approach only involves the uniqueness up to shift of the traveling wave, but does not require any nondegeneracy assumption on the stability of limiting periodic states.Correction to: ``Traveling wave solutions for degenerate nonlinear parabolic equations''https://zbmath.org/1472.350862021-11-25T18:46:10.358925Z"Ichida, Yu"https://zbmath.org/authors/?q=ai:ichida.yu"Sakamoto, Takashi Okuda"https://zbmath.org/authors/?q=ai:sakamoto.takashi-okudaCorrection to the authors' paper [ibid. 6, No. 2, 795--832 (2020; Zbl 1451.35045)].Traveling wave solutions for two species competitive chemotaxis systemshttps://zbmath.org/1472.350872021-11-25T18:46:10.358925Z"Issa, T. B."https://zbmath.org/authors/?q=ai:issa.tahir-bachar"Salako, R. B."https://zbmath.org/authors/?q=ai:salako.rachidi-bolaji"Shen, W."https://zbmath.org/authors/?q=ai:shen.wenxianSummary: In this paper, we consider two species chemotaxis systems with Lotka-Volterra competition reaction terms. Under appropriate conditions on the parameters in such a system, we establish the existence of traveling wave solutions of the system connecting two spatially homogeneous equilibrium solutions with wave speed greater than some critical number \(c^\ast\). We also show the non-existence of such traveling waves with speed less than some critical number \(c_0^\ast\), which is independent of the chemotaxis. Moreover, under suitable hypotheses on the coefficients of the reaction terms, we obtain explicit range for the chemotaxis sensitivity coefficients ensuring \(c^\ast= c_0^\ast\), which implies that the minimum wave speed exists and is not affected by the chemoattractant.\( \varphi^4\) solitons in Kirchhoff wave equationhttps://zbmath.org/1472.350882021-11-25T18:46:10.358925Z"Contoyiannis, Y."https://zbmath.org/authors/?q=ai:contoyiannis.y-f"Papadopoulos, P."https://zbmath.org/authors/?q=ai:papadopoulos.perikles-g"Kampitakis, M."https://zbmath.org/authors/?q=ai:kampitakis.m"Potirakis, S. M."https://zbmath.org/authors/?q=ai:potirakis.stelios-m"Matiadou, N. L."https://zbmath.org/authors/?q=ai:matiadou.niki-linaSummary: We express the Kirchhoff wave equation in terms of classic field theory. This permits us to introduce the spontaneous symmetry breaking phenomenon in the study of linear structures, such as strings in order to investigate the existence of solitons solutions. We find \(\varphi^4\) solitons in the space of spatial gradient of lateral displacement of a string. This helps us detect stable states in deformations of strings.
For the entire collection see [Zbl 1471.34005].Doubly localized two-dimensional rogue waves in the Davey-Stewartson I equationhttps://zbmath.org/1472.350892021-11-25T18:46:10.358925Z"Rao, Jiguang"https://zbmath.org/authors/?q=ai:rao.jiguang"Fokas, Athanassios S."https://zbmath.org/authors/?q=ai:fokas.athanassios-s"He, Jingsong"https://zbmath.org/authors/?q=ai:he.jingsongSummary: Doubly localized two-dimensional rogue waves for the Davey-Stewartson I equation in the background of dark solitons or a constant, are investigated by employing the Kadomtsev-Petviashvili hierarchy reduction method in conjunction with the Hirota's bilinear technique. These two-dimensional rogue waves, described by semi-rational type solutions, illustrate the resonant collisions between lumps or line rogue waves and dark solitons. Due to the resonant collisions, the line rogue waves and lumps in these semi-rational solutions become doubly localized in two-dimensional space and in time. Thus, they are called line segment rogue waves or lump-typed rogue waves. These waves arise from the background of dark solitons, then exist in the background of dark solitons for a very short period of time, and finally completely decay back to the background of dark solitons. In particular circumstances which are characterized by special parametric conditions, the dark solitons in the long wave component of the DSI equation can degenerate into the constant background. In this case, the rogue waves appear and disappear in a constant background.Gevrey regularity of the solutions of the inhomogeneous partial differential equations with a polynomial semilinearityhttps://zbmath.org/1472.350902021-11-25T18:46:10.358925Z"Remy, Pascal"https://zbmath.org/authors/?q=ai:remy.pascalSummary: In this article, we are interested in the Gevrey properties of the formal power series solution in time of the partial differential equations with a polynomial semilinearity and with analytic coefficients at the origin of \(\mathbb{C}^{n+1} \). We prove in particular that the inhomogeneity of the equation and the formal solution are together \(s\)-Gevrey for any \(s\ge s_c\), where \(s_c\) is a nonnegative rational number fully determined by the Newton polygon of the associated linear PDE. In the opposite case \(s<s_c\), we show that the solution is generically \(s_c\)-Gevrey while the inhomogeneity is \(s\)-Gevrey, and we give an explicit example in which the solution is \(s'\)-Gevrey for no \(s'<s_c\).Integral representations of vector functions based on the parametrix of first-order elliptic systemshttps://zbmath.org/1472.350912021-11-25T18:46:10.358925Z"Otelbaev, M."https://zbmath.org/authors/?q=ai:otelbaev.mukhtarbaj-otelbaevich|otelbaev.mukhtarbay"Soldatov, A. P."https://zbmath.org/authors/?q=ai:soldatov.aleksandr-pavlovichSummary: Generalized integrals are introduced with kernels depending on the difference of the arguments taken over a domain and a smooth contour, the boundary of this domain. These kernels arise as parametrixes of first-order elliptic systems with variable coefficients. Using such integrals (with complex density over the domain and real density over the contour), representations of vector functions that are smooth in the closed domain are described. The Fredholmity of the representation obtained in the corresponding Banach spaces is established.Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic bodyhttps://zbmath.org/1472.350922021-11-25T18:46:10.358925Z"Bulíček, Miroslav"https://zbmath.org/authors/?q=ai:bulicek.miroslav"Patel, Victoria"https://zbmath.org/authors/?q=ai:patel.victoria"Şengül, Yasemin"https://zbmath.org/authors/?q=ai:sengul.yasemin"Süli, Endre"https://zbmath.org/authors/?q=ai:suli.endre-eThe authors deal with the periodic with respect to \(x\) initial-boundary value problem describing the viscoelastic bodies exhibiting the strain-limited behaviour
\begin{align*}
&\boldsymbol{u}_{tt}=\text{div}(\mathbb{T})+\boldsymbol{f},\quad \varepsilon(\boldsymbol{u}_{t}+\alpha \boldsymbol{u})= \dfrac{\mathbb{T}}{(1+|\mathbb{T}|^\alpha)^{1/\alpha}}=:F(\mathbb{T})\quad \text{in}\ Q,\\
& \boldsymbol{u}(0,.)=\boldsymbol{u}_0,\ \boldsymbol{u}_t(0,.)=\boldsymbol{v}_0\quad\text{in}\ \Omega,
\end{align*}
where \(\mathbb{T}\) is a symmetric Cauchy stress tensor, \(Q=(0,T)\times\Omega,\ \Omega = (0,1)^d,\ d\ge 2\) and periodic boundary conditions hold. The problem is solved by substitution of monotonic but not bijective function \(\mathbb{T}\mapsto F(\mathbb{T})\) by its regularization \(F_n(\mathbb{T})=F(\mathbb{T})+\dfrac{\mathbb{T}}{n(1+|\mathbb{T}|^{1-1/n})}\). The existence and uniqueness of a weak solution to the original problem is verified.Local existence of solutions for a wave equation with viscoelastic boundary conditionhttps://zbmath.org/1472.350932021-11-25T18:46:10.358925Z"Vo, Giai Giang"https://zbmath.org/authors/?q=ai:vo.giai-giangSummary: The purpose of this paper is to study a wave equation with a viscoelastic boundary condition. Namely, we construct an iterative scheme in order to get a convergent sequence to a local unique weak solution of the problem.Riemann-Hadamard method for one system in three-dimensional spacehttps://zbmath.org/1472.350942021-11-25T18:46:10.358925Z"Mironov, A. N."https://zbmath.org/authors/?q=ai:mironov.aleksei-nikolaevich"Mironova, L. B."https://zbmath.org/authors/?q=ai:mironova.lyubov-borisovnaSummary: For a linear inhomogeneous system of first-order partial differential equations with three independent variables, we prove the existence and uniqueness of a solution of the Darboux problem, which is constructed in terms of the Riemann-Hadamard matrix defined in the paper.Local singular characteristics on \(\mathbb{R}^2\)https://zbmath.org/1472.350952021-11-25T18:46:10.358925Z"Cannarsa, Piermarco"https://zbmath.org/authors/?q=ai:cannarsa.piermarco"Cheng, Wei"https://zbmath.org/authors/?q=ai:cheng.weiSummary: The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on \(\mathbb{R}^2\), two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by \textit{K. Khanin} and \textit{A. Sobolevski} in the paper [Arch. Ration. Mech. Anal. 219, No. 2, 861--885 (2016; Zbl 1333.35201)].Geometric singularities of the solution of the Dirichlet boundary problem for Hamilton-Jacobi equation with a low order of smoothness of the border curvehttps://zbmath.org/1472.350962021-11-25T18:46:10.358925Z"Lebedev, P. D."https://zbmath.org/authors/?q=ai:lebedev.pavel-dmitrievich"Uspenskii, A. A."https://zbmath.org/authors/?q=ai:uspenskii.aleksandr-aleksandrovichSummary: Algorithms for constructing an optimal result function are proposed for a planar time-optimal control problem with a circular velocity vectorogram and a nonconvex compact target set with a smooth boundary. The differential dependencies for smooth segments of a singular set are revealed, which allows them to be considered and constructed in the form of arcs of integral curves. Various types of characteristic points of the boundary of the target set -- so-called pseudo-vertices -- are studied. The necessary conditions for their existence are found and formulas giving the coordinates of the projections of the points of the singular set in their neighborhood are obtained. Examples of time-optimal problems for which numerical construction of the functions of the optimal result and their singular sets are carried out are given. The results are visualized.
For the entire collection see [Zbl 1467.34001].Laplace invariants of differential operatorshttps://zbmath.org/1472.350972021-11-25T18:46:10.358925Z"Hobby, D."https://zbmath.org/authors/?q=ai:hobby.david"Shemyakova, E."https://zbmath.org/authors/?q=ai:shemyakova.ekaterinaSummary: We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that, we investigate general properties of differential invariants of partial differential operators under gauge transformations and introduce a sufficient condition for a set of invariants to be complete. We also give a some mild conditions that guarantee the existence of such a set. The proof is constructive. The method gives many examples of invariants previously known in the literature as well as many new examples including multidimensional.A non-local regularization of the short pulse equationhttps://zbmath.org/1472.350982021-11-25T18:46:10.358925Z"Coclite, Giuseppe Maria"https://zbmath.org/authors/?q=ai:coclite.giuseppe-maria"Di Ruvo, Lorenzo"https://zbmath.org/authors/?q=ai:di-ruvo.lorenzoSummary: The short pulse equation provides a model for the propagation of ultra-short light pulses in silica optical fibers. In this paper, we consider a nonlocal regularization of that equation and prove its well-posedness.On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral conditionhttps://zbmath.org/1472.350992021-11-25T18:46:10.358925Z"Assanova, Anar T."https://zbmath.org/authors/?q=ai:assanova.anar-turmaganbetkyzySummary: Sufficient conditions for the existence and uniqueness of a classical solution to a nonlocal problem for a system of Sobolev-type differential equations with integral condition are established. By introducing a new unknown function, we reduce the considered problem to an equivalent problem consisting of a nonlocal problem for the system of hyperbolic equations of second order with a functional parameter and an integral relation. We propose the algorithm for finding an approximate solution to the investigated problem and prove its convergence.Error analysis of the compliance model for the Signorini problemhttps://zbmath.org/1472.351002021-11-25T18:46:10.358925Z"Cantin, Pierre"https://zbmath.org/authors/?q=ai:cantin.pierre"Hild, Patrick"https://zbmath.org/authors/?q=ai:hild.patrickSummary: The present paper is concerned with a class of penalized Signorini problems also called normal compliance models. These nonlinear models approximate the Signorini problem and are characterized both by a penalty parameter \(\varepsilon\) and by a ``power parameter'' \(\alpha\geq 1\), where \(\alpha=1\) corresponds to the standard penalization. We choose a continuous conforming linear finite element approximation in space dimensions \(d=2,3\) and obtain \(L^2\)-error estimates under various assumptions which are discussed and analyzed.On the dual geometry of Laplacian eigenfunctionshttps://zbmath.org/1472.351012021-11-25T18:46:10.358925Z"Cloninger, Alexander"https://zbmath.org/authors/?q=ai:cloninger.alexander"Steinerberger, Stefan"https://zbmath.org/authors/?q=ai:steinerberger.stefanSummary: We discuss the geometry of Laplacian eigenfunctions \(-\Delta\phi=\lambda\phi\) on compact manifolds \((M, g)\) and combinatorial graphs \(G=(V,E)\). The ``dual'' geometry of Laplacian eigenfunctions is well understood on \(\mathbb{T}^d\) (identified with \(\mathbb{Z}^d)\) and \(\mathbb{R}^n\) (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of ``similarity'' \(\alpha(\phi_\lambda,\phi_\mu)\) between eigenfunctions \(\phi_\lambda\) and \(\phi_\mu\) is given by a global average of local correlations
\[
\alpha(\phi_\lambda,\phi_\mu)^2=||\phi_\lambda\phi_\mu||^{-2}_{L^2}\int_M\left(\int_Mp(t,x,y)(\phi_\lambda(y)-\phi_\lambda(x))(\phi_\mu(y)-\phi_\mu(x))dy\right)^2dx,
\]
where \(p(t,x,y)\) is the classical heat kernel and \(e^{-t\lambda}+e^{-t\mu}=1\). This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.Lipschitz spaces associated to the harmonic oscillatorhttps://zbmath.org/1472.351022021-11-25T18:46:10.358925Z"de León-Contreras, Marta"https://zbmath.org/authors/?q=ai:de-leon-contreras.marta"Torrea, José L."https://zbmath.org/authors/?q=ai:torrea.jose-luisSummary: We define Lipschitz classes adapted to the harmonic ascillator \[\mathcal{H}=-\Delta+|x|^2,\quad x\in\mathbb{R}^n.\] These classes will be defined either through a pointwise condition or through some integral conditions, in this case by using a semigroup approach. We will prove that the different definitions are equivalent. The semigroup approach will allow us to prove regularity properties of some Bessel operators associated to \(\mathcal{H}\).
For the entire collection see [Zbl 1448.65007].The asymptotics for the perfect conductivity problem with stiff \(C^{1,\alpha}\)-inclusionshttps://zbmath.org/1472.351032021-11-25T18:46:10.358925Z"Hao, Xia"https://zbmath.org/authors/?q=ai:hao.xia"Zhao, Zhiwen"https://zbmath.org/authors/?q=ai:zhao.zhiwenConsider bounded domains \(D_1 \subset D \subset \mathbb R^n\) (\(n \geq 2\)) with the boundaries \(\partial D_1\) and \(\partial D\) being of \(C^{1,\alpha}\) (\(0 < \alpha < 1\)). Let \(\nu\) denote the outward unit normal to \(\partial D_1\). The authors discuss the boundary value problem \(\Delta u =0\) in \(D\backslash D_1\), \(u=C\) on \(\partial D_1\), \(\int_{\partial D_1} \frac{\partial u}{\partial \nu} ds =0\), \(u=\phi\) on \(\partial D\) with a given function \(\phi\) and an undetermined constant \(C\). Let \(\varepsilon\) denote the distance between \(\partial D_1\) and \(\partial D\). The authors investigate the behavior of \(u\) and \(\nabla u\) as \(\varepsilon \to 0\) when the curves become touching at one isolated point. The boundary and interior fields are estimated.Dirichlet problem for functions that are harmonic on a two-dimensional nethttps://zbmath.org/1472.351042021-11-25T18:46:10.358925Z"Kovaleva, L. A."https://zbmath.org/authors/?q=ai:kovaleva.l-a"Soldatov, A. P."https://zbmath.org/authors/?q=ai:soldatov.aleksandr-pavlovichSummary: In this paper, we consider the Dirichlet problem for harmonic functions on a two-dimensional complex of a special type. We prove that this problem is a Fredholm problem in the Hölder class and its index is zero.Functional difference equations and eigenfunctions of a Schrödinger operator with \(\delta' -\) interaction on a circular conical surfacehttps://zbmath.org/1472.351052021-11-25T18:46:10.358925Z"Lyalinov, Mikhail A."https://zbmath.org/authors/?q=ai:lyalinov.mikhail-anatolievichSummary: Eigenfunctions and their asymptotic behaviour at large distances for the Laplace operator with singular potential, the support of which is on a circular conical surface in three-dimensional space, are studied. Within the framework of incomplete separation of variables an integral representation of the Kontorovich-Lebedev (KL) type for the eigenfunctions is obtained in terms of solution of an auxiliary functional difference equation with a meromorphic potential. Solutions of the functional difference equation are studied by reducing it to an integral equation with a bounded self-adjoint integral operator. To calculate the leading term of the asymptotics of eigenfunctions, the KL integral representation is transformed to a Sommerfeld-type integral which is well adapted to application of the saddle point technique. Outside a small angular vicinity of the so-called singular directions the asymptotic expression takes on an elementary form of exponent decreasing in distance. However, in an asymptotically small neighbourhood of singular directions, the leading term of the asymptotics also depends on a special function closely related to the function of parabolic cylinder (Weber function).Invariant graphs and spectral type of Schrödinger operatorshttps://zbmath.org/1472.351062021-11-25T18:46:10.358925Z"Avila, Artur"https://zbmath.org/authors/?q=ai:avila.artur"Khanin, Konstantin"https://zbmath.org/authors/?q=ai:khanin.konstantin-m"Leguil, Martin"https://zbmath.org/authors/?q=ai:leguil.martinIn this paper the authors study the spectral properties of the Schrödinger operator in presence of quasi-periodic potentials related to quasi-periodic action minimizing trajectroies for analytic twist maps. They discuss some connections between the Aubry-Mather theory and the spectral theory of the Schrödinger operator. The authors are interested in the 2D twist map setting corresponding to Hamiltonian systems with two degrees of freedom. The main object are the quasi-periodic trajectories for the family of standard type maps on the cylinder, which correspond to minimizer of a suitable Lagrangian and have different properties depending on the coupling constant. The existence of a component of absolutely continuous spectrum when there exists an analytic invariant curve is proved. In the final section of the paper interesting conjectures are given.Nonlinear Schrödinger systems with mixed interactions: locally minimal energy vector solutionshttps://zbmath.org/1472.351072021-11-25T18:46:10.358925Z"Byeon, Jaeyoung"https://zbmath.org/authors/?q=ai:byeon.jaeyoung"Moon, Sang-Hyuck"https://zbmath.org/authors/?q=ai:moon.sang-hyuck"Wang, Zhi-Qiang"https://zbmath.org/authors/?q=ai:wang.zhi-qiangSpectral properties of Landau Hamiltonians with non-local potentialshttps://zbmath.org/1472.351082021-11-25T18:46:10.358925Z"Cárdenas, Esteban"https://zbmath.org/authors/?q=ai:cardenas.esteban"Raikov, Georgi"https://zbmath.org/authors/?q=ai:raikov.georgi-d"Tejeda, Ignacio"https://zbmath.org/authors/?q=ai:tejeda.ignacioSummary: We consider the Landau Hamiltonian \(H_0\), self-adjoint in \(L^2(\mathbb{R}^2)\), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues \(\Lambda_q\), \(q\in\mathbb{Z}_+\). We perturb \(H_0\) by a non-local potential written as a bounded pseudo-differential operator \(\mathrm{Op}^{\mathrm{w}}(\mathcal{V})\) with real-valued Weyl symbol \(\mathcal{V}\), such that \(\mathrm{Op}^{\mathrm{w}}(\mathcal{V})H_0^{-1}\) is compact. We study the spectral properties of the perturbed operator \(H_{\mathcal{V}}=H_0+\mathrm{Op}^{\mathrm{w}}(\mathcal{V})\). First, we construct symbols \(\mathcal{V}\), possessing a suitable symmetry, such that the operator \(H_{\mathcal{V}}\) admits an explicit eigenbasis in \(L^2(\mathbb{R}^2)\), and calculate the corresponding eigenvalues. Moreover, for \(\mathcal{V}\) which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of \(H_{\mathcal{V}}\) adjoining any given \(\Lambda_q\). We find that the effective Hamiltonian in this context is the Toeplitz operator \(\mathcal{T}_q(\mathcal{V})=p_q\mathrm{Op}^{\mathrm{w}}(\mathcal{V}) p_q\), where \(p_q\) is the orthogonal projection onto \(\mathrm{Ker}(H_0-\Lambda_qI\), and investigate its spectral asymptotics.Large energy bubble solutions for Schrödinger equation with supercritical growthhttps://zbmath.org/1472.351092021-11-25T18:46:10.358925Z"Guo, Yuxia"https://zbmath.org/authors/?q=ai:guo.yuxia"Liu, Ting"https://zbmath.org/authors/?q=ai:liu.tingSummary: We consider the following nonlinear Schrödinger equation involving supercritical growth:
\[\begin{cases}
-\Delta u+V(y)u=Q(y)u^{2^*-1+ \varepsilon}\quad \text{in }\mathbb{R}^N,\\
u>0,\quad u\in H^1(\mathbb{R}^N),\end{cases},\tag{\(\ast\)}\]
where \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(N\geq 5\), and \(V(y)\) and \(Q(y)\) are bounded nonnegative functions in \(\mathbb{R}^N \). By using the finite reduction argument and local Pohozaev-type identities, under some suitable assumptions on the functions \(V\) and \(Q\), we prove that for \(\varepsilon>0\) is small enough, problem \((*)\) has large number of bubble solutions whose functional energy is in the order \(\varepsilon^{-\frac{N-4}{(N-2)^2}}.\)Multi-bump standing waves for nonlinear Schrödinger equations with a general nonlinearity: the topological effect of potential wellshttps://zbmath.org/1472.351102021-11-25T18:46:10.358925Z"Jin, Sangdon"https://zbmath.org/authors/?q=ai:jin.sangdonSummary: In this article, we are interested in multi-bump solutions of the singularly perturbed problem
\[-\varepsilon^2\Delta v+V(x)v=f(v)\quad\text{in }\mathbb{R}^N.\]
Extending previous results, we prove the existence of multi-bump solutions for an optimal class of nonlinearities \(f\) satisfying the Berestycki-Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as \(\varepsilon\rightarrow 0\). Examples of potential wells include the following: the union of two compact smooth submanifolds of \(\mathbb{R}^N\) where these two submanifolds meet at the origin and an embedded topological submanifold of \(\mathbb{R}^N \).Ground states for the Schrödinger systems with harmonic potential and combined power-type nonlinearitieshttps://zbmath.org/1472.351112021-11-25T18:46:10.358925Z"Liu, Baiyu"https://zbmath.org/authors/?q=ai:liu.baiyuSummary: We consider a class of coupled nonlinear Schrödinger systems with potential terms and combined power-type nonlinearities. We establish the existence of ground states, by using a variational method. As an application, some symmetry results for ground states of Schrödinger systems with harmonic potential terms are obtained.Localized optical vortex solitons in pair plasmashttps://zbmath.org/1472.351122021-11-25T18:46:10.358925Z"Medina, Luciano"https://zbmath.org/authors/?q=ai:medina.lucianoSummary: The dynamics of short intense electromagnetic pulses propagating in a relativistic pair plasma is governed by a nonlinear Schrödinger equation with a new type of focusing-defocusing saturable nonlinearity. In this context, we provide an existence theory for ring-profiled optical vortex solitons. We prove the existence of both saddle point and minimum type solutions. Via a constrained minimization approach, we prove the existence of solutions where the photon number may be prescribed, and we get the nonexistence of small-photon-number solutions. We also use the constrained minimization to compute the soliton's profile as a function of the photon number and other relevant parameters.Best regularity for a Schrödinger type equation with non smooth data and interpolation spaceshttps://zbmath.org/1472.351132021-11-25T18:46:10.358925Z"Rakotoson, Jean Michel"https://zbmath.org/authors/?q=ai:rakotoson.jean-michelSummary: Given a vector field \(U(x)\) and a nonnegative potential \(V(x)\) on a smooth open bounded set \(\Omega\) of \(\mathbb{R}^n\), we shall discuss some regularity results for the following equation \[-\Delta\omega +U\cdot\nabla\omega+V\omega=f\quad \text{ in }\Omega\tag{0.1}\] whenever \(\delta f\) is a bounded Radon measure with \(\delta(x)\) is the distance between \(x\) and the boundary \(\delta\Omega\).
For the entire collection see [Zbl 1448.65007].Existence and nonexistence of solution for a class of quasilinear Schrödinger equations with critical growthhttps://zbmath.org/1472.351142021-11-25T18:46:10.358925Z"Severo, Uberlandio B."https://zbmath.org/authors/?q=ai:severo.uberlandio-batista"de S. Germano, Diogo"https://zbmath.org/authors/?q=ai:de-s-germano.diogoSummary: In this work, we study the existence and nonexistence of solution for the following class of quasilinear Schrödinger equations:
\[
-\mathrm{div} (g^2(u)\nabla u) + g(u)g'(u)|\nabla u|^2 + V(x)u = f(x,u) + h(x)g(u) \quad \text{in}\quad \mathbb{R}^N,
\]
where \(N\geq 3\), \(g:\mathbb{R}\rightarrow \mathbb{R}_+\) is a continuously differentiable function, \(V(x)\) is a potential that can change sign, the function \(h(x)\) belongs to \(L^{2N/(N+2)}(\mathbb{R}^N)\) and the nonlinearity \(f(x,s)\) is possibly discontinuous and may exhibit critical growth. In order to obtain the nonexistence result, we deduce a Pohozaev identity and the existence of solution is proved by means of a fixed point theorem.Second-order regularity estimates for singular Schrödinger equations on convex domainshttps://zbmath.org/1472.351152021-11-25T18:46:10.358925Z"Tao, Xiangxing"https://zbmath.org/authors/?q=ai:tao.xiangxingSummary: Let \(\Omega \subset \mathbb{R}^n\) be a nonsmooth convex domain and let \(f\) be a distribution in the atomic Hardy space \(H_{a t}^p(\Omega)\); we study the Schrödinger equations \(- \operatorname{div}(A \nabla u) + V u = f\) in \(\Omega\) with the singular potential \(V\) and the nonsmooth coefficient matrix \(A\). We will show the existence of the Green function and establish the \(L^p\) integrability of the second-order derivative of the solution to the Schrödinger equation on \(\Omega\) with the Dirichlet boundary condition for \(n /(n + 1) < p \leq 2\). Some fundamental pointwise estimates for the Green function are also given.Decay and scattering in energy space for the solution of weakly coupled Schrödinger-Choquard and Hartree-Fock equationshttps://zbmath.org/1472.351162021-11-25T18:46:10.358925Z"Tarulli, M."https://zbmath.org/authors/?q=ai:tarulli.mirko"Venkov, G."https://zbmath.org/authors/?q=ai:venkov.georgeSummary: We prove decay with respect to some Lebesgue norms for a class of Schrödinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space for the solutions to the systems of \(N\) defocusing Schrödinger-Choquard equations with mass-energy intercritical nonlinearities in any space dimension and of defocusing Hartree-Fock equations, for any dimension \(d\geq 3\).Existence results for quasilinear Schrödinger equations under a general critical growth termhttps://zbmath.org/1472.351172021-11-25T18:46:10.358925Z"Xue, Yan-Fang"https://zbmath.org/authors/?q=ai:xue.yanfang"Han, Jian-Xin"https://zbmath.org/authors/?q=ai:han.jianxin"Zhu, Xin-Cai"https://zbmath.org/authors/?q=ai:zhu.xincaiSummary: We study the existence of solutions for the following quasilinear Schrödinger equation:
\[
-\Delta u-\Delta (u^2)u=|u|^{2\cdot 2^*-2}u+g(u), \quad x\in \mathbb{R}^N,
\]
where \(N\geq 3\) and \(g\) satisfies very weak growth conditions. The method is to analyze the behavior of solutions for subcritical problems from \textit{M. Colin} and \textit{L. Jeanjean}'s work [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 56, No. 2, 213--226 (2004; Zbl 1035.35038)] and to take the limit as the exponent approaches the critical exponent.Infinitely many solutions for a gauged nonlinear Schrödinger equation with a perturbationhttps://zbmath.org/1472.351182021-11-25T18:46:10.358925Z"Xu, Jiafa"https://zbmath.org/authors/?q=ai:xu.jiafa"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie.5|liu.jie.1|liu.jie|liu.jie.4|liu.jie.7|liu.jie.2|liu.jie.3"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, we use the Fountain theorem under the Cerami condition to study the gauged nonlinear Schrödinger equation with a perturbation in \(\mathbb{R}^2\). Under some appropriate conditions, we obtain the existence of infinitely many high energy solutions for the equation.Some remarks on \(W^{s,p}\) interior elliptic regularity estimateshttps://zbmath.org/1472.351192021-11-25T18:46:10.358925Z"Behzadan, A."https://zbmath.org/authors/?q=ai:behzadan.aliSummary: With the help of certain inequalities which we have established for multiplication in Sobolev-Slobodeckij spaces, we will give a simple proof of a version of elliptic estimate for elliptic operators that belong to a certain general class of operators denoted by \(D^{e,q}_m\). This version of elliptic estimate is not restricted to second order operators and the coefficients are not assumed to be smooth. Our proof is tailored to accommodate fractional Sobolev spaces and our methods are elementary in nature in the sense that they do not directly appeal to the theory of pseudo-differential operators or Littlewood-Paley theory. The elliptic estimate whose proof is presented in this paper has applications in the existence theory of several important PDEs that appear in the study of Einstein constraint equations in general relativity.Hessian estimates for non-divergence form elliptic equations arising from composite materialshttps://zbmath.org/1472.351202021-11-25T18:46:10.358925Z"Dong, Hongjie"https://zbmath.org/authors/?q=ai:dong.hongjie"Xu, Longjuan"https://zbmath.org/authors/?q=ai:xu.longjuanThe authors consider an elliptic operator in non-divergence form and look for interior estimates for the solutions. The equation is settled in a domain containing \(M\) disjoint sub-domains with \(C^{1,\mathrm{Dini}}\) boundaries. In particular the authors assume that the coefficients and the data are allowed to be discontinuous across the interfacial boundaries. They prove piecewise \(C^2\) regularity and a local \(W^{2,\infty}\) estimate for the \(W^{2,1}\) strong solutions in case the coefficients and the data are piecewise Dini continuous in the \(L1-\)mean sense in each subdomain. To this aim they use the Campanato's approach presented in [\textit{S. Campanato}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 17, 175--188 (1963; Zbl 0121.29201); \textit{M. Giaquinta}, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton, NJ: Princeton University Press (1983; Zbl 0516.49003)], showing that the mean oscillation of \(D^2u\) (or \(Du\), or \(u\), respectively) in balls vanishes to certain order as the radii of the balls go to zero. As a byproduct, the authors also obtain a similar result for the adjoint problem.A dynamical system approach to a class of radial weighted fully nonlinear equationshttps://zbmath.org/1472.351212021-11-25T18:46:10.358925Z"Maia, Liliane"https://zbmath.org/authors/?q=ai:maia.liliane-a"Nornberg, Gabrielle"https://zbmath.org/authors/?q=ai:nornberg.gabrielle"Pacella, Filomena"https://zbmath.org/authors/?q=ai:pacella.filomenaSummary: In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator \(\mathcal{M}^-\).The existence of solutions to the nonhomogeneous \(A\)-harmonic equations with variable exponenthttps://zbmath.org/1472.351222021-11-25T18:46:10.358925Z"Wen, Haiyu"https://zbmath.org/authors/?q=ai:wen.haiyuSummary: We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneous \(A\)-harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equation \(d^\star A(x, d \omega) = B(x, d \omega)\) in the weighted variable exponent Sobolev space \(W_d^{p(x)}(\Omega, \Lambda^l, \mu) \).Regularizing effect of two hypotheses on the interplay between coefficients in some Hamilton-Jacobi equationshttps://zbmath.org/1472.351232021-11-25T18:46:10.358925Z"Arcoya, David"https://zbmath.org/authors/?q=ai:arcoya.david"Boccardo, Lucio"https://zbmath.org/authors/?q=ai:boccardo.lucioSummary: We study of the regularizing effect of the interaction between the coefficient of the zero-order term and the lower-order term in quasilinear Dirichlet problems whose model is \[\int_{\Omega}M(x,u)\nabla u\cdot\nabla\varphi+\int_{\Omega}a(x)u\varphi=\int_{\Omega}b(x)|\nabla u|^q\varphi+\int_{\Omega}f(x)\varphi\quad\text{for all } \varphi\in W_0^{1,2}(\Omega)\cap L^{\infty}(\Omega),\]
where \(\Omega\) is a bounded open set of \(\mathbb{R}^N\), \(M(x,s)\) is a Carathéodory matrix on \(\Omega\times\mathbb{R}\) which is elliptic (that is, \(M(x,s)\xi\cdot\xi\geq\alpha|\xi|^2>0\) for every \((x,s,\xi)\in\Omega\times\mathbb{R}\times(\mathbb{R}^N\setminus\{0\}))\) and bounded (that is, \(|M(x,s)|\leq\beta\) for every \((x,s)\in\Omega\times\mathbb{R})\), \(b(x)\in L^{\frac{2}{2-q}}(\Omega)\), \(1<q<2\) and \(0\leq a(x)\in L^1(\Omega)\). We prove the existence of a weak solution \(u\) belonging to \(W_0^{1,2}(\Omega)\) and to \(L^{\infty}(\Omega)\) when either
\[\begin{cases}
b\in L^{\frac{2m}{2-q}}(\Omega)\text{ for some }m>\frac{N}{2} \text{ and}\\
\exists\,Q>0\text{ such that }|f(x)|\leq Qa(x)
\end{cases}\tag{0.1}\]
or
\[\begin{cases}
f\in L^m(\Omega)\text{ for some }m>\frac{N}{2}\text{ and} \\
\exists\,R>0\text{ such that }|b(x)|^{\frac{2}{2-q}}\leq Ra(x).
\end{cases}\tag{0.2}\]
In addition, we also prove the existence for every \(f\in L^1(\Omega)\) and \(b(x)\in L^{\frac{2}{2-q}}(\Omega)\) satisfying both conditions (0.1) and (0.2) jointly.A nonlinear homotopy between two linear Dirichlet problemshttps://zbmath.org/1472.351242021-11-25T18:46:10.358925Z"Boccardo, Lucio"https://zbmath.org/authors/?q=ai:boccardo.lucio"Buccheri, Stefano"https://zbmath.org/authors/?q=ai:buccheri.stefanoSummary: In this paper we focus on the following problem with nonlinear convection term
\[ \begin{cases}
-\text{div}(M(x)\nabla u)= -\text{div}(u|u|^{\theta -1}E(x))+f(x) \quad & \text{in } \Omega ,\\
u (x) = 0 & \text{ on } \partial \Omega ,
\end{cases}\]
where \(\Omega\) is an open bounded domain of \({\mathbb{R}}^N\), with \(N\ge 3\), \(M(x)\) is a uniform elliptic matrix with measurable entries, \( \theta \in (0,1)\), \(E(x)\in (L^r(\Omega ))^N\), with \(r\in (2,N)\), and \(f(x)\in L^m(\Omega )\), with \(m>1\). In particular we study how the relation between the parameters \(\theta\) and \(r\) affects existence and summability of solutions.Global regularity estimates for non-divergence elliptic equations on weighted variable Lebesgue spaceshttps://zbmath.org/1472.351252021-11-25T18:46:10.358925Z"Bui, The Quan"https://zbmath.org/authors/?q=ai:bui.the-quan"Bui, The Anh"https://zbmath.org/authors/?q=ai:the-anh-bui."Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinhConcavity properties of solutions to Robin problemshttps://zbmath.org/1472.351262021-11-25T18:46:10.358925Z"Crasta, Graziano"https://zbmath.org/authors/?q=ai:crasta.graziano"Fragalà, Ilaria"https://zbmath.org/authors/?q=ai:fragala.ilariaThe authors of this article consider two problems for an open uniformly convex bounded domain \(\Omega\) belonging to class \(\mathcal{C}^m\) in \(N\) dimensions with a boundary \(\partial \Omega\) satisfying \(\lfloor m-N/2\rfloor \geq 4\). Specifically, the Robin ground state of \(\Omega\) is considered: \(-\Delta u=\lambda^\beta u\) in \(\Omega\) and \(\partial_\nu u+\beta u=0\) on \(\partial \Omega\) where \(\lambda^\beta\) is the first Robin eigenvalue of \(\Omega\). Moreover, the Robin torsion function of \(\Omega\) is considered: \(-\Delta u=1\) in \(\Omega\) and \(\partial_\nu u+\beta u=0\) on \(\partial \Omega\). It is shown that they are \(\log\)-concave and \(1/2\)-concave, respectively under the assumption that the Robin parameter is greater a critical threshold depending on \(N\), \(m\) as well as on the geometry of the domain; that is, on the diameter and the curvature.Corrigendum to: ``Iterative method for Kirchhoff-carrier type equations and its applications''https://zbmath.org/1472.351272021-11-25T18:46:10.358925Z"Dai, Qiuyi"https://zbmath.org/authors/?q=ai:dai.qiuyiFrom the text: Although the conclusion of Theorem 1.1 in our paper [ibid. 271, 332--342 (2021; Zbl 1454.35095)] is correct, our original proof contains a gap. In the present erratum, we give a new rigorous proof by making use of a so-called invariant set observed
in iterative procedure used in [loc. cit.] and Schaulder fixed point theorem.Uniqueness for quasilinear elliptic problems in a two-component domain with \(L^1\) datahttps://zbmath.org/1472.351282021-11-25T18:46:10.358925Z"Fulgencio, Rheadel G."https://zbmath.org/authors/?q=ai:fulgencio.rheadel-g"Guibé, Olivier"https://zbmath.org/authors/?q=ai:guibe.olivierSummary: In the present paper, we prove the uniqueness of the renormalized solution of the class of quasilinear elliptic problems with \(L^1\) data given by
\[
\begin{cases}
-\operatorname{div}(B(x,u_1)\nabla u_1)=f & \text{ in }\Omega_1, \\
-\operatorname{div}(B(x,u_2)\nabla u_2)=f & \text{ in } \Omega_2, \\
u_1=0 & \text{ on }\partial\Omega, \\ (B(x,u_1)\nabla u_1)\nu_1=(B(x,u_2)\nabla u_2)\nu_1 & \text{ on }\Gamma,\\
(B(x,u_1)\nabla u_1)\nu_1=-h(x)(u_1-u_2) & \text{ on }\Gamma.
\end{cases}
\]
The open sets \(\Omega_1\) and \(\Omega_2\), with \(\Gamma\) as the interface between them, are the two components of the domain \(\Omega\). The data \(f\) is in \(L^1(\Omega)\). In addition to uniform ellipticity, we also prescribe the assumption that the matrix field \(B\) is locally Lipschitz continuous with respect to the second variable.Existence of global steady subsonic Euler flows with collision through 2D infinitely long nozzleshttps://zbmath.org/1472.351292021-11-25T18:46:10.358925Z"Han, Fangyu"https://zbmath.org/authors/?q=ai:han.fangyu"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhong.1|tan.zhongSummary: In this paper, we study the global existence of steady subsonic flows with collision, where the collision is caused by a confluence of two semi-infinitely incoming flows that are nonmiscible steady subsonic irrotational Euler flows come from two different infinitely nozzles. First, we prove that when the total flux of two incoming flows is less that the critical mass flux, there exists a unique global smooth subsonic flow with collision. Meanwhile, the interface between two flows is a smooth free interface, which is determined uniquely by the mass fluxes of incoming flows. Second, by using the blowup argument, we establish the asymptotic behaviors for the stream function. Finally, we prove that the position of free interface can be determined uniquely by the mass fluxes of incoming flows. Moreover, we establish the monotonicity of the relation between the position of free interface and the mass fluxes of incoming flows.Existence of solutions of anisotropic elliptic equations with variable indices of nonlinearity in \(\mathbb{R}^N\)https://zbmath.org/1472.351302021-11-25T18:46:10.358925Z"Kozhevnikova, L. M."https://zbmath.org/authors/?q=ai:kozhevnikova.larisa-mikhailovna"Kamaletdinov, A. Sh."https://zbmath.org/authors/?q=ai:kamaletdinov.a-shSummary: In this paper, we consider a certain class of anisotropic second-order elliptic equations of divergent type with variable indices of nonlinearities. We examine conditions of the solvability in the whole space \(\mathbb{R}^n\), \(n \geq 2\). We prove the existence of solutions without restrictions to the growth rate as \(| x| \rightarrow \infty \).The Dirichlet problem on almost Hermitian manifoldshttps://zbmath.org/1472.351312021-11-25T18:46:10.358925Z"Li, Chang"https://zbmath.org/authors/?q=ai:li.chang"Zheng, Tao"https://zbmath.org/authors/?q=ai:zheng.taoSummary: We prove second-order a priori estimate on the boundary for the Dirichlet problem of a class of fully nonlinear equations on compact almost Hermitian manifolds with smooth boundary. As applications, we solve the Dirichlet problem of the Monge-Ampère type equation and of the degenerate Monge-Ampère equation.Biharmonic problem with Dirichlet and Steklov-type boundary conditions in weighted spaceshttps://zbmath.org/1472.351322021-11-25T18:46:10.358925Z"Matevossian, H. A."https://zbmath.org/authors/?q=ai:matevossian.hovik-a|matevosyan.o-aSummary: The uniqueness of solutions of a biharmonic problem with Dirichlet and Steklov-type boundary conditions in the exterior of a compact set are studied under the assumption that the generalized solution of this problem has a finite Dirichlet integral with a weight \(| x |^a\). Depending on the parameter \(a\), uniqueness (non-uniqueness) theorems are proved and exact formulas for calculating the dimension of the solution space of this biharmonic problem are found.Existence and multiplicity of positive solutions for a critical weight elliptic system in \(\mathbb{R}^N\)https://zbmath.org/1472.351332021-11-25T18:46:10.358925Z"da Silva, João Pablo Pinheiro"https://zbmath.org/authors/?q=ai:da-silva.joao-pablo-pinheiro"de Oliveira, Claudionei Pereira"https://zbmath.org/authors/?q=ai:de-oliveira.claudionei-pereiraExistence of nontrivial solutions for perturbed elliptic system in \(\mathbb{R}^N\)https://zbmath.org/1472.351342021-11-25T18:46:10.358925Z"Jiang, Juan"https://zbmath.org/authors/?q=ai:jiang.juanSummary: We consider the perturbed nonlinear elliptic system \(- \varepsilon^2 \Delta u + V(x) u = K(x) | u |^{2^* - 2} u + H_u(u, v)\), \(x \in \mathbb{R}^N\), \(- \varepsilon^2 \Delta v + V(x) v = K(x) | v |^{2^* - 2} v + H_v(u, v)\), \( x \in \mathbb{R}^N\), where \(N \geq 3\), \(2^* = 2 N /(N - 2)\) is the Sobolev critical exponent. Under proper conditions on \(V\), \(H\), and \(K\), the existence result and multiplicity of the system are obtained by using variational method provided \(\varepsilon\) is small enough.Local boundedness for weak solutions to some quasilinear elliptic systemshttps://zbmath.org/1472.351352021-11-25T18:46:10.358925Z"Leonardi, Salvatore"https://zbmath.org/authors/?q=ai:leonardi.salvatore"Leonetti, Francesco"https://zbmath.org/authors/?q=ai:leonetti.francesco"Pignotti, Cristina"https://zbmath.org/authors/?q=ai:pignotti.cristina"Rocha, Eugenio"https://zbmath.org/authors/?q=ai:rocha.eugenio-a-m"Staicu, Vasile"https://zbmath.org/authors/?q=ai:staicu.vasileSummary: We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi's counterexample. Here we assume a condition on the support of off-diagonal coefficients that ``keeps away'' the counterexample and allows us to prove local boundedness of weak solutions.Sharp estimate for the critical parameters of \(SU(3)\) Toda system with arbitrary singularities. Ihttps://zbmath.org/1472.351362021-11-25T18:46:10.358925Z"Lin, Chang-Shou"https://zbmath.org/authors/?q=ai:lin.chang-shou"Yang, Wen"https://zbmath.org/authors/?q=ai:yang.wenSummary: To obtain the a priori estimate of Toda system, the first step is to determine all the possible local masses of blow up solutions. In this paper we study this problem and improve the main results in [the first author et al., Anal. PDE 8, No. 4, 807--837 (2015; Zbl 1322.35038)]. Our method is based on a recent work by \textit{A. Eremenko} et al. [Ill. J. Math. 58, No. 3, 739--755 (2014; Zbl 1405.30005)].On Schrödinger-Poisson systems involving concave-convex nonlinearities via a novel constraint approachhttps://zbmath.org/1472.351372021-11-25T18:46:10.358925Z"Sun, Juntao"https://zbmath.org/authors/?q=ai:sun.juntao"Wu, Tsung-Fang"https://zbmath.org/authors/?q=ai:wu.tsungfangGround state solutions for planar coupled system involving nonlinear Schrödinger equations with critical exponential growthhttps://zbmath.org/1472.351382021-11-25T18:46:10.358925Z"Wei, Jiuyang"https://zbmath.org/authors/?q=ai:wei.jiuyang"Lin, Xiaoyan"https://zbmath.org/authors/?q=ai:lin.xiaoyan"Tang, Xianhua"https://zbmath.org/authors/?q=ai:tang.xian-huaSummary: We consider the following two coupled nonlinear Schrödinger system:
\[
\begin{cases}
- \Delta u + u = f_1 (x , u) + \lambda (x) v, & x \in \mathbb{R}^2 ,\\
- \Delta v + v = f_2 (x , v) + \lambda (x) u , & x \in \mathbb{R}^2 ,
\end{cases}
\]
where the coupling parameter satisfies \(0 < \lambda (x) \leq \lambda_0 < 1\) and the reactions \(f_1, f_2\) have critical exponential growth in the sense of Trudinger-Moser inequality. Using non-Nehari manifold method together with the Lions's concentration compactness and the Trudinger-Moser inequality, we show that the above system has a Nehari-type ground state solution and a nontrivial solution. Our results improve and extend the previous results.Solvability of an inhomogeneous boundary value problem for steady MHD equationshttps://zbmath.org/1472.351392021-11-25T18:46:10.358925Z"Zhang, Zhibing"https://zbmath.org/authors/?q=ai:zhang.zhibing"Zhao, Chunyi"https://zbmath.org/authors/?q=ai:zhao.chunyiSummary: In this paper, we consider the steady MHD equations with inhomogeneous boundary conditions for the velocity and the tangential component of the magnetic field. Using a new construction of the magnetic lifting, we obtain existence of weak solutions under sharp assumption on boundary data for the magnetic field.On a class of coupled critical Hartree system with deepening potentialhttps://zbmath.org/1472.351402021-11-25T18:46:10.358925Z"Zheng, Yu"https://zbmath.org/authors/?q=ai:zheng.yu.2"Gao, Fashun"https://zbmath.org/authors/?q=ai:gao.fashun"Shen, Zifei"https://zbmath.org/authors/?q=ai:shen.zifei"Yang, Minbo"https://zbmath.org/authors/?q=ai:yang.minboSummary: In this paper, we are interested in the following coupled critical Hartree system
\[
\begin{cases}
-\Delta u + \lambda V_1 (x) u = \alpha_1 u + \beta v + (|x|^{- \mu} \ast |v|^{2_{\mu}^{\ast}}) |u|^{2_{\mu}^{\ast} - 2} u \quad & \text{in } \mathbb{R}^N , \\
-\Delta v + \lambda V_2 (x) v = \beta u + \alpha_2 v + (|x|^{- \mu} \ast |u|^{2_{\mu}^{\ast}}) |v|^{2_{\mu}^{\ast} - 2} v & \text{in } \mathbb{R}^N ,
\end{cases}
\]
where \(\lambda > 0\), \(\beta \geq 0\), \(\alpha_1\), \(\alpha_2 \in \mathbb{R}\), \(0 < \mu < N\), \(N \geq 4\), \(2_{\mu}^{\ast} = (2N - \mu) /(N - 2)\) are the upper critical exponents due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function \(V_1, V_2 \in \mathcal{C}( \mathbb{R}^N, \mathbb{R})\) has potential as well. By using variational methods, we prove the existence of ground state solutions and also characterize the asymptotic behavior of the solutions as the parameter \(\lambda\) goes to infinity. Furthermore, we are able to find the existence of multiple semitrivial weak solutions by the Lusternik-Schnirelmann category theory.Method of scaling spheres for integral and polyharmonic systemshttps://zbmath.org/1472.351412021-11-25T18:46:10.358925Z"Le, Phuong"https://zbmath.org/authors/?q=ai:le.phuong-m|le.phuong-quynhSummary: We establish a method of scaling spheres for the integral system
\[\begin{cases}
u ( x ) = \int_{\mathbb{R}^n} \frac{ | y |^a v^p ( y )}{ | x - y |^{n - \alpha}} d y , & x \in \mathbb{R}^n , \\
v ( x ) = \int_{\mathbb{R}^n} \frac{ | y |^b u^q ( y )}{ | x - y |^{n - \beta}} d y , & x \in \mathbb{R}^n ,
\end{cases}\]
where \(0 < \alpha\), \(\beta < n\), \(a > - \alpha\), \(b > - \beta\) and \(p, q > 0\). By using this method, we obtain a Liouville theorem for nonnegative solutions when \(0 < p \leq \frac{ n + \alpha + 2 a}{ n - \beta}\), \(0 < q \leq \frac{ n + \beta + 2 b}{ n - \alpha}\) and \((p, q) \neq(\frac{ n + \alpha + 2 a}{ n - \beta}, \frac{ n + \beta + 2 b}{ n - \alpha})\). As an application, we derive a Liouville theorem for nonnegative solutions of the polyharmonic Hénon-Hardy system
\[\begin{cases}
( - {\Delta} )^m u ( x ) = | x |^a v^p ( x )\quad & \text{in } \mathbb{R}^n , \\
( - {\Delta} )^l v ( x ) = | x |^b u^q ( x ) & \text{in } \mathbb{R}^n ,
\end{cases}\]
where \(m\) and \(l\) are integers in \((0, \frac{ n}{ 2})\).The Oseen-Frank energy functional on manifoldshttps://zbmath.org/1472.351422021-11-25T18:46:10.358925Z"Hong, Min-Chun"https://zbmath.org/authors/?q=ai:hong.minchunSummary: We observe that for a unit tangent vector field \(u\in TM\) on a 3-dimensional Riemannian manifold \(M\), there is a unique unit cotangent vector field \(A\in T^\ast M\) associated to \(u\) such that we can define the curl of \(u\) by \(dA\). Through a unit cotangent vector field \(A\in T^\ast M\), we define the Oseen-Frank energy functional on 3-dimensional Riemannian manifolds. Moreover, we prove partial regularity of minimizers of the Oseen-Frank energy on 3-dimensional Riemannian manifolds.Compactness and sharp lower bound for a 2D smectics modelhttps://zbmath.org/1472.351432021-11-25T18:46:10.358925Z"Novack, Michael"https://zbmath.org/authors/?q=ai:novack.michael-r"Yan, Xiaodong"https://zbmath.org/authors/?q=ai:yan.xiaodongSummary: We consider a 2D smectics model
\[
E_{\epsilon}(u)=\frac{1}{2}\int_\varOmega\frac{1}{\varepsilon}\left(u_z-\frac{1}{2}u_x^2\right)^2+\varepsilon(u_{xx})^2\text{d}x\,\text{d}z.
\]
For \(\varepsilon_n\rightarrow 0\) and a sequence \(\{u_n\}\) with bounded energies \(E_{\varepsilon_n}(u_n)\), we prove compactness of \(\{\partial_zu_n\}\) in \(L^2\) and \(\{\partial_xu_n\}\) in \(L^q\) for any \(1\le q<p\) under the additional assumption \(\Vert\partial_xu_n\Vert_{L^p}\le C\) for some \(p>6\). We also prove a sharp lower bound on \(E_{\varepsilon}\) when \(\varepsilon\rightarrow 0.\) The sharp bound corresponds to the energy of a 1D ansatz in the transition region.On strongly quasilinear elliptic systems with weak monotonicityhttps://zbmath.org/1472.351442021-11-25T18:46:10.358925Z"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Balaadich, Farah"https://zbmath.org/authors/?q=ai:balaadich.farahSummary: In this paper, we prove existence results in the setting of Sobolev spaces for a strongly quasilinear elliptic system by means of Young measures and mild monotonicity assumptions.Sensitivity of the second order homogenized elasticity tensor to topological microstructural changeshttps://zbmath.org/1472.351452021-11-25T18:46:10.358925Z"Calisti, V."https://zbmath.org/authors/?q=ai:calisti.v"Lebée, A."https://zbmath.org/authors/?q=ai:lebee.arthur"Novotny, A. A."https://zbmath.org/authors/?q=ai:novotny.antonio-andre"Sokolowski, J."https://zbmath.org/authors/?q=ai:sokolowski.janSummary: The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient typehttps://zbmath.org/1472.351462021-11-25T18:46:10.358925Z"Candela, Anna Maria"https://zbmath.org/authors/?q=ai:candela.anna-maria"Salvatore, Addolorata"https://zbmath.org/authors/?q=ai:salvatore.addolorata"Sportelli, Caterina"https://zbmath.org/authors/?q=ai:sportelli.caterinaSummary: The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type
\[\begin{cases}
-\operatorname{div}(A(x,u)|\nabla u|^{p_{1}-2}\nabla u)+\frac{1}{p_1}A_u(x,u)|\nabla u|^{p_1} =G_u( x,u,v) & \text{in }{\Omega} , \\
-\operatorname{div}(B(x,v)|\nabla v|^{p_2-2}\nabla v)+\frac{1}{ p_2}B_v(x,v)|\nabla v|^{p_2} =G_v(x,u,v) &\text{in }{\Omega} , \\ u=v=0 \quad &\text{on }\partial \Omega,
\end{cases}\tag{P}\]
where \(\Omega\subset\mathbb{R}^N\) is an open bounded domain, \(p_1, p_2>1\) and \(A(x,u), B(x,v)\) are \(\mathcal{C}^1 \)-Carathéodory functions on \(\Omega\times\mathbb{R}\) with partial derivatives \(A_u(x,u)\), respectively \(B_v(x,v)\), while \(G_u(x,u,v)\), \(G_v(x,u,v)\) are given Carathéodory maps defined on \(\Omega\times\mathbb{R}\times\mathbb{R}\) which are partial derivatives of a function \(G(x,u,v)\). We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional \(\mathcal{J} \), related to problem (P), admits at least one critical point in the ``right'' Banach space \(X\). Moreover, if \(\mathcal{J}\) is even, then (P) has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami-Palais-Smale condition, a ``good'' decomposition of the Banach space \(X\) and suitable generalizations of the Ambrosetti-Rabinowitz Mountain Pass Theorems.Infinitely many nontrivial solutions of resonant cooperative elliptic systems with superlinear termshttps://zbmath.org/1472.351472021-11-25T18:46:10.358925Z"Chen, Guanwei"https://zbmath.org/authors/?q=ai:chen.guanwei"Ma, Shiwang"https://zbmath.org/authors/?q=ai:ma.shiwangSummary: We study a class of resonant cooperative elliptic systems and replace the Ambrosetti-Rabinowitz superlinear condition with general superlinear conditions. We obtain ground state solutions and infinitely many nontrivial solutions of this system by a generalized Nehari manifold method developed recently by Szulkin and Weth.Existence of bounded solutions for some quasilinear degenerate elliptic systemshttps://zbmath.org/1472.351482021-11-25T18:46:10.358925Z"Di Gironimo, Patrizia"https://zbmath.org/authors/?q=ai:di-gironimo.patrizia"Leonetti, Francesco"https://zbmath.org/authors/?q=ai:leonetti.francesco"Macri, Marta"https://zbmath.org/authors/?q=ai:macri.marta"Petricca, Pier Vincenzo"https://zbmath.org/authors/?q=ai:petricca.pier-vincenzoSummary: We prove the existence of a bounded solution to a quasilinear system of degenerate equations. The main assumption asks the off-diagonal coefficients to have a ``butterfly'' support.Multiple solutions for critical nonhomogeneous elliptic systems in noncontractible domainhttps://zbmath.org/1472.351492021-11-25T18:46:10.358925Z"Duan, Xueliang"https://zbmath.org/authors/?q=ai:duan.xueliang"Wei, Gongming"https://zbmath.org/authors/?q=ai:wei.gongming"Yang, Haitao"https://zbmath.org/authors/?q=ai:yang.haitaoSummary: The paper is concerned with multiple solutions of a nonhomogeneous elliptic system with Sobolev critical exponent over a noncontractible domain, precisely, a smooth bounded annular domain. We prove the existence of four solutions using variational methods and Lusternik-Schnirelmann theory, when the inner hole of the annulus is sufficiently small.A coupled system of \(k\)-Hessian equationshttps://zbmath.org/1472.351502021-11-25T18:46:10.358925Z"Feng, Meiqiang"https://zbmath.org/authors/?q=ai:feng.meiqiang"Zhang, Xuemei"https://zbmath.org/authors/?q=ai:zhang.xuemeiSummary: In this paper, we consider the following system coupled by multiparameter \(k\)-Hessian equations
\[
\begin{cases}
S_k (D^2 u_1 ) = \lambda_1 f_1 (- u_2) & \text{in} \quad \Omega ,\\
S_k (D^2 u_2) = \lambda_2 f_2 (- u_1) & \text{in} \quad \Omega ,\\
u_1 = u_2 = 0 \quad \text{on} \quad \partial \Omega .
\end{cases}
\]
Here, \(\lambda_1\) and \(\lambda_2\) are positive parameters, \(\Omega\) is the unit ball in \(R^n, S_k (D^2 u)\) is the \(k\)-Hessian operator of \(u, \frac{n}{2} < k \leq n\). Applying the eigenvalue theory in cones, several new results are obtained for the existence and multiplicity of nontrivial radial solutions for the above \(k\)-Hessian system. In particular, we study the dependence of the nontrivial radial solution \(u = (u_{\lambda_1}, u_{\lambda_2})\) on the parameter \(\lambda_1\) and \(\lambda_2\). This is probably the first time that a system of equations, especially with fully nonlinear equations, has been studied by applying this technique. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial solutions of the power-type coupled system of \(k\)-Hessian equations, which is new even for the special case \(k=1\) and extends a previous result for the case \(k=n\).Infinitely many solutions for \((p(x),q(x))\)-Laplacian-like systemshttps://zbmath.org/1472.351512021-11-25T18:46:10.358925Z"Heidari, Samira"https://zbmath.org/authors/?q=ai:heidari.samira"Razani, Abdolrahman"https://zbmath.org/authors/?q=ai:razani.abdolrahmanSummary: Variational method has played an important role in solving problems of uniqueness and existence of the nonlinear works as well as analysis. It will also be extremely useful for researchers in all branches of natural sciences and engineers working with non-linear equations economy, optimization, game theory and medicine. Recently, the existence of infinitely many weak solutions for some non-local problems of Kirchhoff type with Dirichlet boundary condition are studied [\textit{M. M. Chaharlang} and the second author, Commun. Korean Math. Soc. 34, No. 1, 155--167 (2019; Zbl 1426.35119)]. Here, a suitable method is presented to treat the elliptic partial derivative equations, especially \((p(x),q(x))\)-Laplacian-like systems. This kind of equations are used in the study of fluid flow, diffusive transport akin to diffusion, rheology, probability, electrical networks, etc. Here, the existence of infinitely many weak solutions for some boundary value problems involving the \((p(x),q(x))\)-Laplacian-like operators is proved. The method is based on variational methods and critical point theory.\(L^r\)-Helmholtz-Weyl decomposition for three dimensional exterior domainshttps://zbmath.org/1472.351522021-11-25T18:46:10.358925Z"Hieber, Matthias"https://zbmath.org/authors/?q=ai:hieber.matthias"Kozono, Hideo"https://zbmath.org/authors/?q=ai:kozono.hideo"Seyfert, Anton"https://zbmath.org/authors/?q=ai:seyfert.anton"Shimizu, Senjo"https://zbmath.org/authors/?q=ai:shimizu.senjo"Yanagisawa, Taku"https://zbmath.org/authors/?q=ai:yanagisawa.takuSummary: In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the \(L^r\)-setting for \(1 < r < \infty \). In fact, given an \(L^r\)-vector field \(\mathbf{u}\), there exist \(\mathbf{h} \in X_{\operatorname{har}}^r(\Omega)\), \(\mathbf{w} \in \dot{H}^{1 , r} ( \Omega )^3\) with \(\operatorname{div} \mathbf{w} = 0\) and \(p \in \dot{H}^{1 , r}(\Omega)\) such that \(\mathbf{u}\) may be decomposed uniquely as
\[\mathbf{u} = \mathbf{h} + \text{rot} \mathbf{w} + \nabla p .\]
If for the given \(L^r\)-vector field \(\mathbf{u}\), its harmonic part \(\mathbf{h}\) is chosen from \(V_{\operatorname{har}}^r(\Omega)\), then a decomposition similar to the above one is established, too. However, its uniqueness holds in this case only for the case \(1 < r < 3\). The proof given relies on an \(L^r\)-variational inequality allowing to construct \(\mathbf{w} \in \dot{H}^{1 , r} ( \Omega)^3\) and \(p \in \dot{H}^{1 , r}(\Omega)\) for given \(\mathbf{u} \in L^r ( \Omega )^3\) as weak solutions to certain elliptic boundary value problems.Multiplicity of positive solutions for a second-order elliptic system of Kirchhoff typehttps://zbmath.org/1472.351532021-11-25T18:46:10.358925Z"Khademloo, S."https://zbmath.org/authors/?q=ai:khademloo.somayeh|khademloo.somaye"Valipour, E."https://zbmath.org/authors/?q=ai:valipour.ezat"Babakhani, A."https://zbmath.org/authors/?q=ai:babakhani.aydin|babakhani.azizollah|babakhani.aliSummary: We study elliptic problems of Kirchhoff type in \(\Omega \subset \mathbb{R}^N\) (\(N \geq 3\)). Using variational tools, we establish the existence of at least two nontrivial and nonnegative solutions.Corrigendum to: ``A Morse-Smale index theorem for indefinite elliptic systems and bifurcation''https://zbmath.org/1472.351542021-11-25T18:46:10.358925Z"Portaluri, Alessandro"https://zbmath.org/authors/?q=ai:portaluri.alessandro"Waterstraat, Nils"https://zbmath.org/authors/?q=ai:waterstraat.nilsSummary: We discussed in a previous paper [ibid. 258, No. 5, 1715--1748 (2015; Zbl 1310.35111)] elliptic systems of partial differential equations on star-shaped domains and introduced the notions of conjugate radius and bifurcation radius. We proved that every bifurcation radius is a conjugate radius, and believed to have shown by an example that on the other hand not every conjugate radius is a bifurcation radius. This note reveals that our previous example was wrong, but it also introduces an improved example that shows the assertion that we claimed before.Existence results for some nonlinear elliptic equations via topological degree methodshttps://zbmath.org/1472.351552021-11-25T18:46:10.358925Z"Abbassi, Adil"https://zbmath.org/authors/?q=ai:abbassi.adil"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakir"Kassidi, Abderrazak"https://zbmath.org/authors/?q=ai:kassidi.abderrazakSummary: This article is devoted to study the existence of weak solutions to a Dirichlet boundary value problem related to the following nonlinear elliptic equation
\[
-\operatorname{div}(a(x,u, \nabla u)) - \lambda g(x,u, \nabla u)=b(x)|u|^{q-2}u,
\]
where \(-\operatorname{div}(a(x,u, \nabla u))\) is a Leray-Lions operator acting from \(W_0^{1,p}(\varOmega ,w)\) to its dual \(W^{-1,p^{\prime}}(\varOmega ,w^{\ast})\). On the nonlinear term \(g(x,s, \eta )\), we only assume the growth condition on \(\eta \). Our approach is based on the topological degree introduced by Berkovits.A compact embedding result for anisotropic Sobolev spaces associated to a strip-like domain and some applicationshttps://zbmath.org/1472.351562021-11-25T18:46:10.358925Z"Alves, Claudianor"https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Molica Bisci, Giovanni"https://zbmath.org/authors/?q=ai:molica-bisci.giovanniThe authors propose to use variational and topological methods together with the principle of symmetric criticality. Precisely, they first establish a compactness embedding result of anisotropic Sobolev spaces, then obtain the existence of infinitely many cylindrically symmetric weak solutions to a Neumann problem involving the \(p(x,y)\)-Laplace differential operator and an oscillating nonlinearity.Existence of solutions for a class of Kirchhoff-type equation via Young measureshttps://zbmath.org/1472.351572021-11-25T18:46:10.358925Z"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Balaadich, Farah"https://zbmath.org/authors/?q=ai:balaadich.farahSummary: In this article, our aim is to obtain weak solutions for a class of Kirchhoff-type problems
\[
-M\left(\int_\Omega A(x,\nabla u)dx\right)\operatorname{div}\, a(x,\nabla u)=f(x,u)\quad\text{in }\Omega,
\]
by using the theory of Young measures.Three solutions for a nonlocal fractional \(p\)-Kirchhoff type elliptic systemhttps://zbmath.org/1472.351582021-11-25T18:46:10.358925Z"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Benkirane, Abdelmoujib"https://zbmath.org/authors/?q=ai:benkirane.abdelmoujib"Boumazourh, Athmane"https://zbmath.org/authors/?q=ai:boumazourh.athmane"Srati, Mohammed"https://zbmath.org/authors/?q=ai:srati.mohammedSummary: In this paper, we study the existence of three weak solutions for a Kirchhoff-type elliptic system involving nonlocal fractional \(p\)-Laplacian with homogeneous Dirichlet boundary conditions. The approach is based on the three critical points theorem and some variational methods.Some existence results for a nonlocal non-isotropic problemhttps://zbmath.org/1472.351592021-11-25T18:46:10.358925Z"Bentifour, Rachid"https://zbmath.org/authors/?q=ai:bentifour.rachid"El-Hadi Miri, Sofiane"https://zbmath.org/authors/?q=ai:miri.sofiane-el-hadiSummary: In this paper we deal with the following problem
\[
\begin{cases}
-\sum\limits_{i=1}^N\left[\left(a+b\int\limits_{\Omega}\vert \partial_iu\vert^{p_i}dx\right)\partial_i(\vert\partial_iu\vert^{p_i-2}\partial_iu)\right]=\frac{f(x)}{u^\gamma}\pm g(x)u^{q-1} &\text{ in }\Omega, \\
u\geq 0 &\text{ in }\Omega, \\
u=0 &\text{ on }\partial\Omega,
\end{cases}
\]
where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^N\). We will assume without loss of generality that \(1\leq p_1\leq p_2\leq\dots\leq p_N\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^m(\Omega)\), \(1<q<\overline{p}^{\ast}\), \(a>0,b>0\) and \(0<\gamma<1\).Second-order PDEs in four dimensions with half-flat conformal structurehttps://zbmath.org/1472.351602021-11-25T18:46:10.358925Z"Berjawi, S."https://zbmath.org/authors/?q=ai:berjawi.s"Ferapontov, E. V."https://zbmath.org/authors/?q=ai:ferapontov.evgeny-vladimirovich"Kruglikov, B."https://zbmath.org/authors/?q=ai:kruglikov.boris-s"Novikov, V."https://zbmath.org/authors/?q=ai:novikov.vladimir-sSummary: We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge-Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge-Ampère type. Some partial classification results of Monge-Ampère equations in four dimensions with half-flat conformal structure are also obtained.Optimal control of the radiation heat exchange equations for multi-component mediahttps://zbmath.org/1472.351612021-11-25T18:46:10.358925Z"Chebotarev, A. Yu."https://zbmath.org/authors/?q=ai:chebotarev.alexander-yurievichSummary: An analysis of optimal control problems for nonlinear elliptic equations modeling complex heat transfer with Fresnel conjugation conditions on the discontinuity surfaces of the refractive index is presented. Conditions for the solvability of extremal problems and the nondegeneracy of the optimality system are obtained. For the control problem with boundary observation, the bang-bang property is set.Corrigendum to: ``Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or \(L^1\) data''https://zbmath.org/1472.351622021-11-25T18:46:10.358925Z"Chlebicka, Iwona"https://zbmath.org/authors/?q=ai:chlebicka.iwona"Giannetti, Flavia"https://zbmath.org/authors/?q=ai:giannetti.flavia"Zatorska-Goldstein, Anna"https://zbmath.org/authors/?q=ai:zatorska-goldstein.annaSummary: The authors would like to correct an error in the proof of uniqueness in their paper [ibid. 479, No. 1, 185--213 (2019; Zbl 1433.35086)].Some global results for a class of homogeneous nonlocal eigenvalue problemshttps://zbmath.org/1472.351632021-11-25T18:46:10.358925Z"Dai, Guowei"https://zbmath.org/authors/?q=ai:dai.guoweiRemovable singularities of solutions of the symmetric minimal surface equationhttps://zbmath.org/1472.351642021-11-25T18:46:10.358925Z"Dierkes, Ulrich"https://zbmath.org/authors/?q=ai:dierkes.ulrichSummary: We show that compact sets of \((n-1)\)-dimensional Hausdorff measure zero constitute removable singularities for weak solutions of the singular and symmetric minimal surface equation. These results comprise minimal surfaces in hyperbolic space and ``heavy'' minimal surfaces in gravitational fields.The cone Moser-Trudinger inequalities and applicationshttps://zbmath.org/1472.351652021-11-25T18:46:10.358925Z"Fang, Fei"https://zbmath.org/authors/?q=ai:fang.fei"Ji, Chao"https://zbmath.org/authors/?q=ai:ji.chaoSummary: In this paper, we first study the cone Moser-Trudinger inequalities and their best exponents on both bounded and unbounded domains \(\mathbb{R}_+^2\). Then, using the cone Moser-Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation
\[
\begin{cases}
-\Delta_{\mathbb{B}}u=f(x,u), & \text{ in }x\in\operatorname{int}(\mathbb{B}), \\
u=0, & \text{ on }\partial\mathbb{B},
\end{cases}
\]
where \(\Delta_{\mathbb{B}}\) is an elliptic operator with conical degeneration on the boundary \(x_1=0\), and the nonlinear term \(f\) has the subcritical exponential growth or the critical exponential growth.Positive solutions of the prescribed mean curvature equation with exponential critical growthhttps://zbmath.org/1472.351662021-11-25T18:46:10.358925Z"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcher"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-dGiven a bounded domain \(\Omega \subset \mathbb{R}^2\) with smooth boundary \(\partial\Omega\), the authors study the prescribed mean curvature equation given by
\begin{align*}
-\text{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=f(u) \quad\text{in }\Omega, \quad u=0 \quad\text{on }\partial\Omega,
\end{align*}
where \(f\colon\mathbb{R}\to\mathbb{R}\) is a superlinear continuous function with critical exponential growth. Based on an auxiliary problem along with the Nehari manifold by using Moser's iteration method and Stampacchia's estimates, the existence of a positive solution of the problem above is shown.Infinitely many solutions for a new class of Schrödinger-Kirchhoff type equations in \(\mathbb{R}^N\) involving the fractional \(p\)-Laplacianhttps://zbmath.org/1472.351672021-11-25T18:46:10.358925Z"Hamdani, Mohamed Karim"https://zbmath.org/authors/?q=ai:hamdani.mohamed-karim"Chung, Nguyen Thanh"https://zbmath.org/authors/?q=ai:nguyen-thanh-chung."Bayrami-Aminlouee, Masoud"https://zbmath.org/authors/?q=ai:bayrami-aminlouee.masoudSummary: This paper deals with the existence of infinitely many solutions for a new class of Schrödinger-Kirchhoff type equations of the form
\[
M \left( [u]_{s,p}^p + \int_{\mathbb{R}^N} V(x)|u|^p \, dx \right) \Big[ (-\Delta )_p^s u + V(x)|u|^{p-2}u \Big] = \lambda h(x)|u|^{q-2}u + f(x,u), \; x \in \mathbb{R}^N,
\]
where
\[
[u]_{s,p}^p := \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^p}{|x-y|^{ N+sp}} dx dy,
\]
\(s \in (0,1)\), \(N>sp\), \(p \geq 2\), \(M(t) = a -bt^{\gamma -1}\), \(t \geq 0\), \(1< \gamma < \frac{p^{\ast}_s}{p}\) with \(p^{\ast}_s = \frac{Np}{N-sp}\), \(a- \frac{b}{\gamma }>0\) with \(a,b \in \mathbb{R}_0^+ := [0, \infty )\), \(\lambda \) is a parameter, \(q \in (1,p)\), \((- \Delta )_p^s\) is the fractional \(p\)-Laplace operator, \(V : \mathbb{R}^N \rightarrow \mathbb{R}^+ := (0, \infty )\) is a potential function, \(h\) is a sign-changing weight function and \(f\) is a continuous function satisfying the Ambrosetti-Rabinowitz condition or not. To our best knowledge, the results here are the first contributions to the study of fractional Schrödinger-Kirchhoff type equations in which the Kirchhoff functions may be sign-changing and degenerate.New class of sixth-order nonhomogeneous \(p(x)\)-Kirchhoff problems with sign-changing weight functionshttps://zbmath.org/1472.351682021-11-25T18:46:10.358925Z"Hamdani, Mohamed Karim"https://zbmath.org/authors/?q=ai:hamdani.mohamed-karim"Chung, Nguyen Thanh"https://zbmath.org/authors/?q=ai:nguyen-thanh-chung."Repovš, Dušan D."https://zbmath.org/authors/?q=ai:repovs.dusan-dSummary: In this paper, we prove the existence of multiple solutions for the following sixth-order \(p(x)\)-Kirchhoff-type problem
\[
\begin{cases}
-M\left( \int\limits_{\mathit{\Omega}} \frac{1}{p(x)}|\nabla\mathit{\Delta} u|^{p(x)}dx\right)\mathit{\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) &\text{in}\quad\mathit{\Omega}, \\
u = \mathit{\Delta} u = \mathit{\Delta}^2 u = 0, & \text{on}\quad \partial\mathit{\Omega},
\end{cases}
\]
where \(\mathit{\Omega} \subset \mathbb{R}^N\) is a smooth bounded domain, \(N > 3, \mathit{\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div}\Big(\mathit{\Delta}(|\nabla\mathit{\Delta} u|^{p(x)-2}\nabla\mathit{\Delta} u)\Big)\) is the \(p(x)\)-triharmonic operator, \(p, q, r \in C(\overline{\mathit{\Omega}})\), \(1 < p(x) < \dfrac N3\) for all \(x \in \overline{\mathit{\Omega}}, M(s) = a-bs^\gamma, a, b, \gamma > 0, \lambda > 0, g : \mathit{\Omega} \times \mathbb{R} \rightarrow \mathbb{R}\) is a nonnegative continuous function while \(f, h : \mathit{\Omega} \times \mathbb{R} \rightarrow \mathbb{R}\) are sign-changing continuous functions in \(\mathit{\Omega}\). To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order \(p(x)\)-Kirchhoff type problems with sign changing Kirchhoff functions.Ground state solutions for a class of fractional Schrödinger-Poisson system with critical growth and vanishing potentialshttps://zbmath.org/1472.351692021-11-25T18:46:10.358925Z"Meng, Yuxi"https://zbmath.org/authors/?q=ai:meng.yuxi"Zhang, Xinrui"https://zbmath.org/authors/?q=ai:zhang.xinrui"He, Xiaoming"https://zbmath.org/authors/?q=ai:he.xiaoming.1|he.xiaomingSummary: In this paper, we study the fractional Schrödinger-Poisson system
\[
\begin{cases}
(-\Delta)^su+V(x)u+K(x)\phi|u|^{q-2}u=h(x)f(u)+|u|^{2^{\ast}_s-2}u, & \text{ in
}\mathbb{R}^3,\\
(-\Delta)^t\phi=K(x)|u|^q, & \text{ in }\mathbb{R}^3,
\end{cases}
\]
where \(s,t\in (0,1)\), \(3<4s<3+2t\), \(q\in (1,2^*_s/2)\) are real numbers, \((- \Delta)^s\) stands for the fractional Laplacian operator, \(2^*_s:=\frac{6}{3-2s}\) is the fractional critical Sobolev exponent, \(K,V\) and \(h\) are non-negative potentials and \(V,h\) may be vanish at infinity. \(f\) is a \(C^1\)-function satisfying suitable growth assumptions. We show that the above fractional Schrödinger-Poisson system has a positive and a sign-changing least energy solution via variational methods.Strict \(2\)-convexity of convex solutions to the quadratic Hessian equationhttps://zbmath.org/1472.351702021-11-25T18:46:10.358925Z"Mooney, Connor"https://zbmath.org/authors/?q=ai:mooney.connorThe author considers convex viscosity solutions to the quadratic Hessian inequality \[ \sigma_2(D^2u) \ge 1, \tag{1} \] where the function \(\sigma_k\) on \(\mathrm{Sym}_{n\times n}\) denotes the \(k^{th}\) symmetric polynomial of the eigenvalues. It is elliptic on the cone \[\Gamma_k := \big\{M \in \mathrm{Sym}_{n\times n} : \sigma_l(M) > 0 \;\, \mbox{for each} \;\, 1 \le l \le k\big\},\] and has convex level sets in \(\Gamma_k\). Futhermore we say that a function \(u \in C^2(\Omega)\) is \(k\)-convex if \(D^2u \in \overline{\Gamma_k}\). Given a nonnegative function \(f \in C(\Omega)\), we say that a function \(u \in C(\Omega)\) is a viscosity solution of \[\sigma_k(D^2u) \ge (\le) f\] if, whenever a \(k\)-convex function \(\varphi \in C^2(\Omega)\) touches \(u\) from above (below) at a point \(x_0 \in \Omega\), we have \[\sigma_k(D^2\varphi(x_0)) \ge (\le) f(x_0).\]
The main results are the following theorems.
Theorem 1.1. Let \(u\) be a convex viscosity solution to (1) in \(\Omega \subset \mathbb{R}^n\), and let \(L\) be a supporting linear function to \(u\) in \(\Omega\). Then \[ \dim\{u = L\} \le n - 2. \]
With a compactness argument the author also proves a universal modulus of strict 2-convexity (Proposition 4.1) and obtains:
Theorem 1.2. Let \(u\) be a convex viscosity solution of \(\sigma_2(D^2u) = 1\) in \(B_1 \subset \mathbb{R}^n\). Then \(u\) is smooth, and \[ |D^2u(0)| \le C(n,\|u\|_{L^\infty(B_1)}). \]Qualitative analysis of solutions for the \(p\)-Laplacian hyperbolic equation with logarithmic nonlinearityhttps://zbmath.org/1472.351712021-11-25T18:46:10.358925Z"Pişkin, Erhan"https://zbmath.org/authors/?q=ai:piskin.erhan"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Irkil, Nazli"https://zbmath.org/authors/?q=ai:irkil.nazliSummary: In this paper, the \(p\)-Laplacian hyperbolic type equation with logarithmic nonlinearity and weak damping term are considered, where the global existence of solutions by using the potential well method is discussed. Furthermore, the growth and the decay estimates of solutions for the problem are studied.A singular elliptic problem involving fractional \(p\)-Laplacian and a discontinuous critical nonlinearityhttps://zbmath.org/1472.351722021-11-25T18:46:10.358925Z"Saoudi, Kamel"https://zbmath.org/authors/?q=ai:saoudi.kamel"Panda, Akasmika"https://zbmath.org/authors/?q=ai:panda.akasmika"Choudhuri, Debajyoti"https://zbmath.org/authors/?q=ai:choudhuri.debajyotiSummary: The purpose of this article is to prove the existence of solution to a nonlinear nonlocal elliptic problem with a singularity and a discontinuous critical nonlinearity, which is given as \((- \Delta)_p^s u =\mu g(x, u) + \frac{\lambda}{u^\gamma} + H(u - \alpha) u^{p_s^* - 1}\) in \(\Omega\), \(u> 0\) in \(\Omega\), with the zero Dirichlet boundary condition. Here, \(\Omega \subset \mathbb{R}^N\) is
a bounded domain with Lipschitz boundary, \(s \in (0, 1)\), \(2 < p < \frac{N}{s}\), \(\gamma \in (0, 1)\), \(\lambda, \mu > 0\), \(\alpha \geq 0\) is real, \(p_s^* = \frac{N p}{N - s p}\) is the fractional critical Sobolev exponent, and \(H\) is the Heaviside function, i.e., \(H(a) = 0\) if \(a \leq 0\) and \(H(a) = 1\) if \(a > 0\). Under suitable assumptions on the function \(g\), the existence of solution to the problem has been established. Furthermore, it will be shown that as \(\alpha \rightarrow 0^+\), the sequence of solutions of the problem for each such \(\alpha\) converges to a solution of the problem for which \(\alpha = 0\).Ground state and nodal solutions for critical Schrödinger-Kirchhoff-type Laplacian problemshttps://zbmath.org/1472.351732021-11-25T18:46:10.358925Z"Zhang, Huabo"https://zbmath.org/authors/?q=ai:zhang.huaboSummary: In this paper, we are interested in the existence of ground state nodal solutions for the following Schrödinger-Kirchhoff-type Laplacian problems:
\[
-M \left( \int_{\mathbb{R}^3} |\nabla u|^2 \mathrm{d}x \right) \Delta u+V(x)u=|u|^4 u+ k f(u), \; x \in \mathbb{R}^3 ,
\]
where \(M(t)=a+bt^{\gamma}\) with \(0< \gamma <2, a,b>0\) and the nonlinear function \(f \in C(\mathbb{R}, \mathbb{R})\). By the nodal Nehari manifold method, for each \(b>0\), we obtain a least energy nodal solution \(u_b\) and a ground state solution \(v_b\) of this problems when \(k \gg 1\). Our results improve and extend the known results of the usual case \(\gamma =1\) in the sense that a more wider range of \(\gamma \) is covered.On the Klein-Gordon-Maxwell system with critical exponential growth in \(\mathbb{R}^2\)https://zbmath.org/1472.351742021-11-25T18:46:10.358925Z"Chen, Sitong"https://zbmath.org/authors/?q=ai:chen.sitong"Lin, Xiaoyan"https://zbmath.org/authors/?q=ai:lin.xiaoyan"Tang, Xianhua"https://zbmath.org/authors/?q=ai:tang.xian-huaSummary: This paper is concerned with the following Klein-Gordon-Maxwell system:
\[
\begin{cases}
- \Delta u + V (x) u - ( 2 \omega + \phi ) \phi u = f(u) , \quad & x \in \mathbb{R}^2 , \\
\Delta \phi = (\omega + \phi ) u^2 , & x \in \mathbb{R}^2 ,
\end{cases}
\]
where \(\omega > 0\) is a constant, \(u, \phi : \mathbb{R}^2 \to \mathbb{R}, V \in \mathcal{C}( \mathbb{R}^2, \mathbb{R})\) and \(f \in \mathcal{C}(\mathbb{R}, \mathbb{R})\) obeys exponential critical growth in the sense of the Trudinger-Moser inequality. We give some new sufficient conditions on \(f\), specifically related to exponential growth, to obtain the existence of nontrivial solutions. Furthermore, we prove a nonexistence result with a Pohozaev-type argument which also provides important information to get the above existence results. In particular, some new analytical approaches and estimates are used to overcome the difficulty arising from the critical growth of Trudinger-Moser type.Half-space Gaussian symmetrization: applications to semilinear elliptic problemshttps://zbmath.org/1472.351752021-11-25T18:46:10.358925Z"Díaz, J. I."https://zbmath.org/authors/?q=ai:diaz.jesus-ildelfonso|diaz.jesus-idelfonso|diaz-diaz.jesus-ildefonso"Feo, F."https://zbmath.org/authors/?q=ai:feo.filomena"Posteraro, M. R."https://zbmath.org/authors/?q=ai:posteraro.maria-rosariaSummary: We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in \(\mathbb{R}^N\) without growth restrictions at infinity. A comparison result in terms of the half-space Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type.Mixed fractional Sobolev spaces and elliptic PDEs with singular integral boundary datahttps://zbmath.org/1472.351762021-11-25T18:46:10.358925Z"Merker, Jochen"https://zbmath.org/authors/?q=ai:merker.jochenSummary: In this article, we show how to use mixed fractional Sobolev spaces to prove existence of very weak solutions to singular semilinear elliptic PDEs subject to singular integral Neumann boundary conditions. Particularly, we obtain solutions with infinite normal derivatives on the boundary, ie, solutions with large derivatives.Existence of solutions for some non-Fredholm integro-differential equations with the bi-Laplacianhttps://zbmath.org/1472.351772021-11-25T18:46:10.358925Z"Vougalter, Vitali"https://zbmath.org/authors/?q=ai:vougalter.vitali"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: We prove the existence of solutions for some semilinear elliptic equations in the appropriate \(H^4\) spaces using the fixed-point technique where the elliptic equation contains fourth-order differential operators with and without Fredholm property, generalizing the previous results.An indefinite quasilinear elliptic problem with weights in anisotropic spaceshttps://zbmath.org/1472.351782021-11-25T18:46:10.358925Z"Abreu, Emerson"https://zbmath.org/authors/?q=ai:abreu.emerson-a-m"Felix, Diego"https://zbmath.org/authors/?q=ai:felix.diego-dias"Medeiros, Everaldo"https://zbmath.org/authors/?q=ai:medeiros.everaldo-sSummary: In this paper we consider existence, nonexistence and multiplicity of solutions for a class of indefinite quasilinear elliptic problems in the upper half-space involving weights in anisotropic Lebesgue spaces. One of our basic tools consists in a Hardy type inequality proved in the present paper that allows us to establish Sobolev embeddings into Lebesgue spaces with weights in anisotropic Lebesgue spaces.On singular quasilinear elliptic equations with data measureshttps://zbmath.org/1472.351792021-11-25T18:46:10.358925Z"Alaa, Nour Eddine"https://zbmath.org/authors/?q=ai:alaa.noureddine"Aqel, Fatima"https://zbmath.org/authors/?q=ai:aqel.fatima-al-zahra"Taourirte, Laila"https://zbmath.org/authors/?q=ai:taourirte.lailaSummary: The aim of this work is to study a quasilinear elliptic equation with singular nonlinearity and data measure. Existence and non-existence results are obtained under necessary or sufficient conditions on the data, where the main ingredient is the isoperimetric inequality. Finally, uniqueness results for weak solutions are given.Multiplicity results for critical Kirchhoff problems involving concave-convex nonlinearitieshttps://zbmath.org/1472.351802021-11-25T18:46:10.358925Z"Benmansour, Safia"https://zbmath.org/authors/?q=ai:benmansour.safia"Matallah, Atika"https://zbmath.org/authors/?q=ai:matallah.atika"Litimein, Sara"https://zbmath.org/authors/?q=ai:litimein.saraSummary: This paper is concerned with the existence and the multiplicity of solutions for elliptic Kirchhoff problems containing critical Sobolev exponent and defined on a regular bounded domain in \(\mathbb{R}^3\). The results are obtained via the Nehari manifold and Ekeland's variational principle.Quasilinear problems without the Ambrosetti-Rabinowitz conditionhttps://zbmath.org/1472.351812021-11-25T18:46:10.358925Z"Candela, Anna Maria"https://zbmath.org/authors/?q=ai:candela.anna-maria"Fragnelli, Genni"https://zbmath.org/authors/?q=ai:fragnelli.genni"Mugnai, Dimitri"https://zbmath.org/authors/?q=ai:mugnai.dimitriSummary: We show the existence of nontrivial solutions for a class of quasilinear problems in which the governing operators depend on the unknown function. By using a suitable variational setting and a weak version of the Cerami-Palais-Smale condition, we establish the desired result without assuming that the nonlinear source satisfies the Ambrosetti-Rabinowitz condition.Infinitely many solutions of a quasilinear elliptic equation with nonlinearity oscillating close to zerohttps://zbmath.org/1472.351822021-11-25T18:46:10.358925Z"de Araujo, Anderson L. A."https://zbmath.org/authors/?q=ai:de-araujo.anderson-luis-albuquerque"Abreu, Rafael dos Reis"https://zbmath.org/authors/?q=ai:abreu.rafael-dos-reisSummary: We find infinitely many solutions close to zero of the quasilinear elliptic equation \(-\operatorname{div}\left (\phi (|\nabla u|^2)\nabla u\right )=f(x,u)\) in \(\Omega\) with Dirichlet's boundary condition, where \(\Omega\) is a smooth bounded domain in \({\mathbb{R}}^N\), \(N\ge 1\) and \(f:\Omega \times{\mathbb{R}} \rightarrow{\mathbb{R}}\) is an unbounded continuous function with oscillatory behavior near the origin.Multiple symmetric results for quasilinear elliptic systems involving singular potentials and critical Sobolev exponents in \(\mathbb{R}^N\)https://zbmath.org/1472.351832021-11-25T18:46:10.358925Z"Deng, Zhiying"https://zbmath.org/authors/?q=ai:deng.zhiying"Huang, Yisheng"https://zbmath.org/authors/?q=ai:huang.yishengSummary: This paper deals with a class of quasilinear elliptic systems involving singular potentials and critical Sobolev exponents in \(\mathbb{R}^N\). By using the symmetric criticality principle of Palais and variational methods, we prove several existence and multiplicity results of \(G\)-symmetric solutions under certain appropriate hypotheses on the potentials and parameters.Existence result for a Dirichlet problem governed by nonlinear degenerate elliptic equation in weighted Sobolev spaceshttps://zbmath.org/1472.351842021-11-25T18:46:10.358925Z"El Ouaarabi, Mohamed"https://zbmath.org/authors/?q=ai:el-ouaarabi.mohamed"Abbassi, Adil"https://zbmath.org/authors/?q=ai:abbassi.adil"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakirSummary: In this paper, we prove the existence and uniqueness of solution to a Dirichlet boundary value problems for the following nonlinear degenerate elliptic equation
\[
-\mbox{div} \Big [ \omega_1 \mathcal{A} (x, \nabla u)+ \omega_2 \mathcal{B} (x,u, \nabla u)\Big ] + \omega_3 b(x,u)+ \omega_4 |u|^{p-2}u=f(x),
\]
where \(\omega_1, \omega_2, \omega_3\) and \(\omega_4\) are weight functions, \(\mathcal{A} : \varOmega \times \mathbb{R}^n \longrightarrow \mathbb{R}^n\) , \(\mathcal{B} : \varOmega \times \mathbb{R} \times \mathbb{R}^n \longrightarrow \mathbb{R}^n\) , \(b : \varOmega \times \mathbb{R} \longrightarrow \mathbb{R}\) are Caratéodory functions that satisfy some conditions and the right-hand side term \(f\) belongs to \(L^1(\varOmega )\).Bound state solutions of Choquard equations with a nonlocal operatorhttps://zbmath.org/1472.351852021-11-25T18:46:10.358925Z"Guo, Lun"https://zbmath.org/authors/?q=ai:guo.lun"Li, Qi"https://zbmath.org/authors/?q=ai:li.qi|li.qi.1Summary: In this paper, we study the following Choquard equation with Kirchhoff operator
\[
- \left(a + b \int_{\mathbb{R}^N} |\nabla u|^2 \right) \Delta u + V(x)u = (I_{\alpha} \ast |u|^{2_{\alpha}^{\ast}}) |u|^{2_{\alpha}^{\ast} - 2} u, \quad x \in \mathbb{R}^N, \tag{\(0.1\)}
\]
where \(a \geq 0\), \(b > 0\), \(\alpha \in (0, N)\), \(2_{\alpha}^{\ast} = \frac{N + \alpha}{N - 2}\) is the critical exponent respect to Hardy-Littlewood-Sobolev inequality, and \(V(x) \in L^{\frac{N}{2}}(\mathbb{R}^N)\) is a given nonnegative function. By using the classical linking theorem and global compactness theorem, we prove that equation (0.1) has at least one bound state solution if \(\| V \|_{L^{\frac{N}{2}}}\) is small. More intriguingly, our result covers a novel feature of Kirchhoff problems, which is the fact that the parameter \(a\) can be zero.Two solutions for a class of singular Kirchhoff-type problems with Hardy-Sobolev critical exponent IIhttps://zbmath.org/1472.351862021-11-25T18:46:10.358925Z"Liao, Jia-Feng"https://zbmath.org/authors/?q=ai:liao.jiafengSummary: In this article, we devote ourselves to investigate the following singular Kirchhoff-type equation:
\[
\begin{cases}
-( a + b \int_{\Omega} | \nabla u |^2 dx) \Delta u = \frac{ u^{5 - 2 s}}{ | x |^s} + \frac{ \lambda}{ | x |^\beta u^\gamma}, & x \in \Omega ,\\
u > 0, & x \in \Omega ,\\
u = 0, & x \in \partial \Omega ,
\end{cases}
\]
where \(\Omega \subset \mathbb{R}^3\) is a bounded domain with smooth boundary \(\partial \Omega\), \(0 \in \Omega\), \(a \geq 0\), \(b, \lambda > 0\), \(0 < \gamma\) , \(s < 1\), and \(0 \leq \beta < \frac{ 5 + \gamma}{ 2}\). By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.Asymptotic behavior of integral functionals for a two-parameter singularly perturbed nonlinear traction problemhttps://zbmath.org/1472.351872021-11-25T18:46:10.358925Z"Falconi, Riccardo"https://zbmath.org/authors/?q=ai:falconi.riccardo"Luzzini, Paolo"https://zbmath.org/authors/?q=ai:luzzini.paolo"Musolino, Paolo"https://zbmath.org/authors/?q=ai:musolino.paoloSummary: We consider a nonlinear traction boundary value problem for the Lamé equations in an unbounded periodically perforated domain. The edges lengths of the periodicity cell are proportional to a positive parameter \(\delta \), whereas the relative size of the holes is determined by a second positive parameter \(\varepsilon \). Under suitable assumptions on the nonlinearity, there exists a family of solutions \(\{ u (\varepsilon , \delta , \cdot ) \}_{( \varepsilon , \delta ) \in ]0, \varepsilon^{\prime}[ \times ]0, \delta^{\prime}[}\). We analyze the asymptotic behavior of two integral functionals associated to such a family of solutions when the perturbation parameter pair \((\varepsilon , \delta )\) is close to the degenerate value \((0, 0)\).Existence of the eigenvalues for the cone degenerate \(p\)-Laplacianhttps://zbmath.org/1472.351882021-11-25T18:46:10.358925Z"Chen, Hua"https://zbmath.org/authors/?q=ai:chen.hua"Wei, Yawei"https://zbmath.org/authors/?q=ai:wei.yaweiSummary: The present paper is concerned with the eigenvalue problem for cone degenerate \(p\)-Laplacian. First the authors introduce the corresponding weighted Sobolev s-paces with important inequalities and embedding properties. Then by adapting Lusternik-Schnirelman theory, they prove the existence of infinity many eigenvalues and eigenfunctions. Finally, the asymptotic behavior of the eigenvalues is given.A revisit of elliptic variational-hemivariational inequalitieshttps://zbmath.org/1472.351892021-11-25T18:46:10.358925Z"Han, Weimin"https://zbmath.org/authors/?q=ai:han.weiminSummary: In this paper, we provide an alternative approach to establish the solution existence and uniqueness for elliptic variational-hemivariational inequalities. The new approach is based on elementary results from functional analysis, and thus removes the need of the notion of pseudomonotonicity and the dependence on surjectivity results for pseudomonotone operators. This makes the theory of elliptic variational-hemivariational inequalities more accessible to applied mathematicians and engineers. In addition, equivalent minimization principles are further explored for particular elliptic variational-hemivariational inequalities. Representative examples from contact mechanics are discussed to illustrate application of the theoretical results.\(\Sigma\)-shaped bifurcation curveshttps://zbmath.org/1472.351902021-11-25T18:46:10.358925Z"Acharya, A."https://zbmath.org/authors/?q=ai:acharya.arjun-prasad|acharya.akshay-s|acharya.ankit-s|acharya.arup|acharya.anirudh|acharya.amit|acharya.ayan|acharya.anish|acharya.a-n|acharya.aditya|acharya.arnab|acharya.avidit|acharya.anurag"Fonseka, N."https://zbmath.org/authors/?q=ai:fonseka.nalin"Quiroa, J."https://zbmath.org/authors/?q=ai:quiroa.j"Shivaji, R."https://zbmath.org/authors/?q=ai:shivaji.ratnasinghamSummary: We study positive solutions to the steady state reaction diffusion equation of the form:
\[
\begin{cases}
-\Delta u=\lambda f(u);\, \Omega\\
\frac{\partial u}{\partial\eta}+\sqrt{\lambda}u=0;\,\partial\Omega
\end{cases}
\]
where \(\lambda>0\) is a positive parameter, \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) when \(N>1\) (with smooth boundary \(\partial\Omega)\) or \(\Omega=(0,1)\), and \(\frac{\partial u}{\partial \eta}\) is the outward normal derivative of \(u\). Here \(f(s)=ms+g(s)\) where \(m\geq 0\) (constant) and \(g\in C^2[0, r)\cap C[0,\infty)\) for some \(r>0\). Further, we assume that \(g\) is increasing, sublinear at infinity, \(g(0)=0\), \(g'(0)=1\) and \(g''(0)>0\). In particular, we discuss the existence of multiple positive solutions for certain ranges of \(\lambda\) leading to the occurrence of \(\Sigma\)-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.Multiplicity of solutions for Neumann problems for semilinear elliptic equationshttps://zbmath.org/1472.351912021-11-25T18:46:10.358925Z"An, Yu-Cheng"https://zbmath.org/authors/?q=ai:an.yucheng"Suo, Hong-Min"https://zbmath.org/authors/?q=ai:suo.hong-minSummary: Using the minimax methods in critical point theory, we study the multiplicity of solutions for a class of Neumann problems in the case near resonance. The results improve and generalize some of the corresponding existing results.Supercritical problems with concave and convex nonlinearities in \(\mathbb{R}^N\)https://zbmath.org/1472.351922021-11-25T18:46:10.358925Z"do Ó, João Marcos"https://zbmath.org/authors/?q=ai:do-o.joao-m-bezerra"Mishra, Pawan Kumar"https://zbmath.org/authors/?q=ai:mishra.pawan-kumar"Moameni, Abbas"https://zbmath.org/authors/?q=ai:moameni.abbasConstruction of solutions for Hénon-type equation with critical growthhttps://zbmath.org/1472.351932021-11-25T18:46:10.358925Z"Guo, Yuxia"https://zbmath.org/authors/?q=ai:guo.yuxia"Liu, Ting"https://zbmath.org/authors/?q=ai:liu.tingSummary: We consider the following Hénon-type problem with critical growth:
\[\begin{cases}
-Delta u = K (|y'|,y'')u^{2^*-1},\quad u>0 & \text{in }B_1,\\ u =0 \quad &\text{on }\partial B_1,
\end{cases}\tag{H}\]
where \(2^*=\frac{2N}{N-2}\), \(N\geq 5\), \(B_1\) is the unit sphere in \(\mathbb{R}^N \), \(y=(y',y'')\in\mathbb{R}^2\times\mathbb{R}^{N-2}\), \(r=|y'|\) and \(K(y)=K(r,y'')\in C^{2 }(B_1)\) is a bounded non-negative function. By using a finite reduction argument and local Pohozaev-type identities, we prove that if \(N\geq 5\) and \(K(r,y'')\) has a stable critical point \(y_0=(r_0,y_0'')\in\partial B_1 \), then the above problem has infinitely many solutions, whose energy can be arbitrarily large.Influence of an \(L^p\)-perturbation on Hardy-Sobolev inequality with singularity a curvehttps://zbmath.org/1472.351942021-11-25T18:46:10.358925Z"Ijaodoro, Idowu Esther"https://zbmath.org/authors/?q=ai:ijaodoro.idowu-esther"Thiam, El Hadji Abdoulaye"https://zbmath.org/authors/?q=ai:thiam.el-hadji-abdoulayeSummary: We consider a bounded domain \(\Omega\) of \(\mathbb{R}^N\), \(N \geq 3\), \(h\) and \(b\) continuous functions on \(\Omega\). Let \(\Gamma\) be a closed curve contained in \(\Omega\). We study existence of positive solutions \(u \in H^1_0(\Omega)\) to the perturbed Hardy-Sobolev equation:
\[
-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1}\text{ in }\Omega,
\]
where \(2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}\) is the critical Hardy-Sobolev exponent, \(\sigma\in [0,2)\), \(0<\delta<\frac{4}{N-2}\) and \(\rho_{\Gamma}\) is the distance function to \(\Gamma\). We show that the existence of minimizers does not depend on the local geometry of \(\Gamma\) nor on the potential \(h\). For \(N=3\), the existence of ground-state solution may depends on the trace of the regular part of the Green function of \(-\Delta+h\) and or on \(b\). This is due to the perturbative term of order \(1+\delta\).Positive radial symmetric solutions for a class of elliptic problems with critical exponent and \(-1\) growthhttps://zbmath.org/1472.351952021-11-25T18:46:10.358925Z"Lei, Chun-Yu"https://zbmath.org/authors/?q=ai:lei.chunyu"Liao, Jia-Feng"https://zbmath.org/authors/?q=ai:liao.jiafengSummary: In this paper, we consider a class of semilinear elliptic equation with critical exponent and \(-1\) growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.An elliptic equation with indefinite nonlinearities and exponential critical growth in \(\mathbb{R}^2\)https://zbmath.org/1472.351962021-11-25T18:46:10.358925Z"Medeiros, Everaldo S."https://zbmath.org/authors/?q=ai:medeiros.everaldo-s"Severo, Uberlandio B."https://zbmath.org/authors/?q=ai:severo.uberlandio-batista"Silva, Elves A. B."https://zbmath.org/authors/?q=ai:silva.elves-a-bThe authors study a semilinear elliptic equation \[ -\Delta u=\lambda u+W(x)f(u)\text{ in }\Omega,\text{ on }\partial\Omega, \] on a bounded, smooth domain \(\Omega\subset\mathbb R^2\). The weight function \(W\) is sign-changing and continuous on \(\Omega\), and the nonlinearity \(f\) is continuous on \(\mathbb R\) with critical exponential growth of Trudinger-Moser type. The authors state sufficient conditions on \(f\) and \(W\) guarantying that the bifurcation from the trivial solution set is on the right of the first eigenvalue, and that the branch of positive solutions has a turning point, so they characterise the range of values of the parameter \(\lambda\) for which the problem has one, none, or at least two positive solutions.
It is an extension for the case \(N=2\), of a known work of [\textit{S. Alama} and \textit{G. Tarantello}, Calc. Var. Partial Differ. Equ. 1, No. 4, 439--475 (1993; Zbl 0809.35022)].Remarks on: ``Existence result for an elliptic equation involving critical exponent in three dimensional domains''https://zbmath.org/1472.351972021-11-25T18:46:10.358925Z"Sharaf, Khadijah"https://zbmath.org/authors/?q=ai:sharaf.khadijah-abdullah|sharaf.khadijahSummary: We consider a nonlinear partial differential equation of Yamabe-type. In [\textit{Z. Boucheche}, Complex Var. Elliptic Equ. 64, No. 4, 649--675 (2019; Zbl 1416.35117)], it has been proved that the problem admits a solution under the assumption that the gradient of the associated variational functional is lower bounded by a positive constant. The purpose of this paper is to provide a counter-example to such a result.An asymptotic monotonicity formula for minimizers of elliptic systems of Allen-Cahn type and the Liouville propertyhttps://zbmath.org/1472.351982021-11-25T18:46:10.358925Z"Sourdis, Christos"https://zbmath.org/authors/?q=ai:sourdis.christosAuthor's abstract: We prove an asymptotic monotonicity formula for bounded, globally minimizing solutions (in the sense of Morse) to a class of semilinear elliptic systems of the form \(\Delta u=W_u(u)\), \(x\in\mathbb R^n\), \(n\geq 2\) with \(W: \mathbb R^m\to\mathbb R\), \(m\geq 1\), nonnegative and vanishing at exactly one point (at least in the closure of the image of the considered solution \(u\)). As an application, we can prove a Liouville type theorem under various assumptions.On uniqueness of solutions to the boundary value problems on the Sierpiński gaskethttps://zbmath.org/1472.351992021-11-25T18:46:10.358925Z"Stegliński, Robert"https://zbmath.org/authors/?q=ai:steglinski.robertSummary: Using the monotonicity methods, we obtain conditions for the existence of the unique weak solution of Dirichlet problem
\[
\begin{cases}
\Delta u(x)+a(x)u(x)=f(x,u(x))\quad x\in V\backslash V_0\\
u_{|V_0}=0,
\end{cases}
\]
considered on the Sierpiński gasket. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are also given. We also consider a problem with parameters and we prove the theorem about continuous dependence on parameters.Multiple results to some biharmonic problemshttps://zbmath.org/1472.352002021-11-25T18:46:10.358925Z"Tang, Xingdong"https://zbmath.org/authors/?q=ai:tang.xingdong"Zhang, Jihui"https://zbmath.org/authors/?q=ai:zhang.jihuiSummary: We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.Multiple solutions to elliptic equations on \(\mathbb{R}^N\) with combined nonlinearitieshttps://zbmath.org/1472.352012021-11-25T18:46:10.358925Z"Yang, Miaomiao"https://zbmath.org/authors/?q=ai:yang.miaomiao"Li, Anran"https://zbmath.org/authors/?q=ai:li.anranSummary: In this paper, we are concerned with the multiplicity of nontrivial radial solutions for the following elliptic equations \((P)_\lambda\: -\Delta u+V(x)u=\lambda Q(x)|u|^{q-2}u+Q(x)f(u)\), \(x\in\mathbb R^N\); \(u(x)\to0\), as \(|x|\rightarrow+\infty\), where \(1<q< 2\), \(0<\lambda\in\mathbb R\), \(N\geq3\), \(V\), and \(Q\) are radial positive functions, which can be vanishing or coercive at infinity, and \(f\) is asymptotically linear at infinity.Fractional \(p\)-Laplacian problem with indefinite weight in \(\mathbb{R}^N\): eigenvalues and existencehttps://zbmath.org/1472.352022021-11-25T18:46:10.358925Z"Cui, Na"https://zbmath.org/authors/?q=ai:cui.na"Sun, Hong-Rui"https://zbmath.org/authors/?q=ai:sun.hongruiSummary: In this paper, we first study the fractional \(p\)-Laplacian eigenvalue problem with indefinite weight
\[
(- \Delta_p)^s u = \lambda g (x) |u|^{p - 2} u \quad \text{in}\quad \mathbb{R}^N
\]
and prove that the problem exists a sequence of eigenvalues that converges to infinity and that the first eigenvalue is simple. Based on these results, we then consider the existence of infinitely many solutions for a class of indefinite weight problem with concave and convex nonlinearities.\(q\)-moment estimates for the singular \(p\)-Laplace equation and applicationshttps://zbmath.org/1472.352032021-11-25T18:46:10.358925Z"Drapeau, Samuel"https://zbmath.org/authors/?q=ai:drapeau.samuel"Yin, Liming"https://zbmath.org/authors/?q=ai:yin.limingSummary: We provide \(q\)-moment estimates on annuli for weak solutions of the singular \(p\)-Laplace equation where \(p\) and \(q\) are conjugates. We derive \(q\)-uniform integrability for some critical parameter range. As applications, we derive a mass conservation as well as a weak convergence result for a larger critical parameter range. Concerning the latter point, we further provide a rate of convergence of order \(t^{q-1}\) of the solution in the \(q\)-Wasserstein distance.Nonexistence of variational minimizers related to a quasilinear singular problem in metric measure spaceshttps://zbmath.org/1472.352042021-11-25T18:46:10.358925Z"Garain, Prashanta"https://zbmath.org/authors/?q=ai:garain.prashanta"Kinnunen, Juha"https://zbmath.org/authors/?q=ai:kinnunen.juhaThe goal of this article is to show that there are cases where the equation
\[
-\Delta_p u=-\operatorname{div}(|\nabla|^{p-2}\nabla u)=f(u)
\]
does not have a positive solution. The idea is to consider a singular problem, that is an example where \(f\) blows up near zero.
The novelty of the paper is that the nonexistence is studied in quite general metric measure spaces. Hence, not exactly the above equation is studied, but an integral version of it with solutions in Sobolev spaces. In the metric setting, this makes it possible to use Newtonian spaces as Sobolev spaces. As \(f\), the authors choose the function given by \(u\mapsto u^{-\delta}\).
The main part of the proof is a lemma that provides an energy estimate for a positive variational minimizer. This is then used to show that such a minimizer must vanish, a contradiction to the existence of a positive minimizer.Standing waves for a class of fractional \(p\)-Laplacian equations with a general critical nonlinearityhttps://zbmath.org/1472.352052021-11-25T18:46:10.358925Z"Lou, Qing-Jun"https://zbmath.org/authors/?q=ai:lou.qingjun"Mao, An-Min"https://zbmath.org/authors/?q=ai:mao.anmin"You, Jin"https://zbmath.org/authors/?q=ai:you.jinSummary: In this paper, we concern with the following fractional \(p\)-Laplacian equation with critical Sobolev exponent
\[
\begin{cases}
\varepsilon^{ps} \{ - \Delta_p^s u + V (x) |u|^{p - 2} u = \lambda f (x) |u|^{q - 2} u + |u|^{p_s^{\ast} - 2} u \quad \text{in } \mathbb{R}^N, \\
u \in W^{s , p} (\mathbb{R}^N), \quad u > 0,
\end{cases}
\]
where \(\varepsilon > 0\) is a small parameter, \( \lambda > 0\), \(N\) is a positive integer, and \(N > ps\) with \(s \in (0, 1)\) fixed, \(1 < q \leq p\), \(p_s^{\ast} := Np / (N - ps)\).
Since the nonlinearity \(h(x, u) := \lambda f(x) |u|^{q - 2} u + |u|^{p_s^{\ast} - 2} u\) does not satisfy the following Ambrosetti-Rabinowitz condition:
\[
0 < \mu H (x,u) := \mu \int_0^u h(x,t)dt \leq h(x,u)u, \quad x \in \mathbb{R}^N, 0 \neq u \in \mathbb{R},
\]
with \(\mu > p\), it is difficult to obtain the boundedness of Palais-Smale sequence, which is important to prove the existence of positive solutions. In order to overcome the above difficulty, we introduce a penalization method of fractional \(p\)-Laplacian type.On \((p(x), q(x))\)-Laplace equations in \(\mathbb{R}^N\) without Ambrosetti-Rabinowitz conditionhttps://zbmath.org/1472.352062021-11-25T18:46:10.358925Z"Nastasi, Antonella"https://zbmath.org/authors/?q=ai:nastasi.antonellaSummary: In the present work, we consider a \((p(x), q(x))\)-elliptic equation describing the behavior of a double-phase anisotropic problem which has relevance in electrorheological fluid applications. The analysis leads to the existence of weak (nonnegative) solutions in the special case of potential terms with critical frequency and a superlinear reaction term. In order to prove the existence result, we combine critical point theory of mountain pass type with related topological and variational methods. Basically, the approach is variational, but we do not impose any Ambrosetti-Rabinowitz type condition for the superlinearity of the reaction. More specifically, we apply the Euler-Lagrange functional approach to the variational formulation of the above-mentioned model problem. We note that we work in the whole space \(\mathbb{R}^N\) and so we have to consider non-compact embeddings. This aspect constitutes an additional difficulty in our study.Multiple solutions for a modified quasilinear Schrödinger elliptic equation with a nonsquare diffusion termhttps://zbmath.org/1472.352072021-11-25T18:46:10.358925Z"Zhang, Xinguang"https://zbmath.org/authors/?q=ai:zhang.xinguang"Liu, Lishan"https://zbmath.org/authors/?q=ai:liu.lishan"Wu, Yonghong"https://zbmath.org/authors/?q=ai:wu.yonghong.1"Wiwatanapataphee, Benchawan"https://zbmath.org/authors/?q=ai:wiwatanapataphee.benchawanSummary: In this paper, we establish the results of multiple solutions for a class of modified nonlinear Schrödinger equation involving the \(p\)-Laplacian. The main tools used for analysis are the critical points theorems by Ricceri and the dual approach.Asymptotic behavior of density in the boundary-driven exclusion process on the Sierpinski gaskethttps://zbmath.org/1472.352082021-11-25T18:46:10.358925Z"Chen, Joe P."https://zbmath.org/authors/?q=ai:chen.joe-p-j|chen.joe-p"Gonçalves, Patrícia"https://zbmath.org/authors/?q=ai:goncalves.patricia-cSummary: We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.Spectral heat content for Lévy processeshttps://zbmath.org/1472.352092021-11-25T18:46:10.358925Z"Grzywny, Tomasz"https://zbmath.org/authors/?q=ai:grzywny.tomasz"Park, Hyunchul"https://zbmath.org/authors/?q=ai:park.hyunchul"Song, Renming"https://zbmath.org/authors/?q=ai:song.renmingSummary: In this paper we study the spectral heat content for various Lévy processes. We establish the small time asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in \(\mathbb R^d, d \geq 1\). We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in \(\mathbb R\) with respect to symmetric Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally, we establish that the small time asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.Fokas method for linear boundary value problems involving mixed spatial derivativeshttps://zbmath.org/1472.352102021-11-25T18:46:10.358925Z"Batal, A."https://zbmath.org/authors/?q=ai:batal.ahmet"Fokas, A. S."https://zbmath.org/authors/?q=ai:fokas.athanassios-s"Özsarı, T."https://zbmath.org/authors/?q=ai:ozsari.turkerSummary: We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.On correctness of a mixed problem for the heat equation with the mixed derivative in the boundary conditionhttps://zbmath.org/1472.352112021-11-25T18:46:10.358925Z"Kapustin, N."https://zbmath.org/authors/?q=ai:kapustin.n-yu|kapustin.nikolay"Kholomeeva, A."https://zbmath.org/authors/?q=ai:kholomeeva.a-a|kholomeeva.annaSummary: We consider an initial-boundary value problem for the heat equation with an inhomogeneous initial condition and boundary conditions. One of the boundary conditions contains a mixed derivative. When solving this problem by the method of separation of variables, a spectral problem arises. A system of eigenfunctions of this spectral problem and a biorthogonally conjugate system, are constructed explicitly. Also we obtain an asymptotic formula for the eigenvalues. In this paper we formulate theorems on the properties of the system of eigenfunctions of the spectral problem and a theorem about representing the solution of the initial initial-boundary value problem in the form of a Fourier series in the system of eigenfunctions. Thus, the existence of a solution is shown if the initial condition belongs to the Holder class. However, it has been shown that the solution is not unique. We show that additional condition guarantees the uniqueness of the solution. The unique solution of this problem is also obtained in the article.Generalized solvability of a parabolic model describing transfer processes in domains with thin inclusionshttps://zbmath.org/1472.352122021-11-25T18:46:10.358925Z"Tymchyshyn, I. B."https://zbmath.org/authors/?q=ai:tymchyshyn.i-b"Nomirovskii, D. A."https://zbmath.org/authors/?q=ai:nomirovskij.d-aSummary: We study a system of first-order partial differential equations with generalized functions as coefficients that describes the heat and mass transfer process in domains with thin inclusions. It is proved that the operator of the problem is continuous and injective, and a theorem on the existence and uniqueness of a generalized solution is also established.Non-local problems with integral displacement for highorder parabolic equationshttps://zbmath.org/1472.352132021-11-25T18:46:10.358925Z"Kozhanov, Aleksandr Ivanovich"https://zbmath.org/authors/?q=ai:kozhanov.aleksandr-ivanovich"Dyuzheva, Aleksandra Vladimirovna"https://zbmath.org/authors/?q=ai:dyuzheva.aleksandra-vladimirovnaSummary: The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation). Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.Global existence of solutions to the Cauchy problem of a two dimensional attraction-repulsion chemotaxis systemhttps://zbmath.org/1472.352142021-11-25T18:46:10.358925Z"Shi, Renkun"https://zbmath.org/authors/?q=ai:shi.renkun"You, Guoqiao"https://zbmath.org/authors/?q=ai:you.guoqiaoThe paper considers the attraction-repulsion chemotaxis system
\[
\begin{cases} u_t-\Delta u=-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\\
v_t+\beta v-\Delta v=\alpha u,\\
\delta w-\Delta w=\gamma u, \quad x\in \mathbb{R}^2,\ \, t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x) ,\quad x\in \mathbb{R}^2. \end{cases}\tag{EQ}
\]
Here \(u\) denotes the cell density, \(v\) denotes the chemoattractant density and \(w\) denotes the chmorepellent density. The positive numbers \(\chi,\xi,\alpha,\beta,\gamma,\delta\) describe various biological parameters involved.
The main result of the paper is, that if one of the following two constraints hold,
a) \(\chi \alpha-\xi\gamma\leq 0\);
b) \(\chi\alpha-\xi \gamma>0\) and \(||u_0||_{L^1}<\frac{8\pi}{\chi\alpha-\xi \gamma}\),
then the system (EQ) has a unique global smooth solution.
The main difficulty in the analysis comes from the fact that the Lyapunov functional constructed by [\textit{H.-Y. Jin} and \textit{Z.-A. Wang}, J. Differ. Equations 260, No. 1, 162--196 (2016; Zbl 1323.35001)] to handle the bounded domain case is not effective on the whole plane. To overcome this difficulty, the authors constructed a modified free energy functional
\begin{multline*}
\mathcal{E}(u,v,w)= \\
\int_{\mathbb{R}^2}(1+u)\ln(1+u) dx+\frac{\chi}{2\alpha}\int_{\mathbb{R}^2}(\beta v^2+|\nabla v|^2)dx+\frac{\xi}{2\gamma}\int_{\mathbb{R}^2}(\delta w^2+|\nabla w|^2)dx-\chi\int_{\mathbb{R}^2}uv dx.
\end{multline*}
By carefully analyzing the differential inequalities involving \(\mathcal{E}\), the authors are able to derive a-priori bounds and apply the Moser-Alikakos iteration to complete the argument.Existence, uniqueness and positivity on a free-boundary high order diffusion cooperative systemhttps://zbmath.org/1472.352152021-11-25T18:46:10.358925Z"Palencia, José Luis Díaz"https://zbmath.org/authors/?q=ai:palencia.jose-luis-diaz"Redondo, Antonio Naranjo"https://zbmath.org/authors/?q=ai:redondo.antonio-naranjoSummary: We study existence, uniqueness and positivity conditions for a cooperative system formulated with a high order diffusion. In addition, we show and characterize the instabilities due to the high order diffusion for which a positivity and a comparison principle hold after rescaling. Instabilities shall be understood as the oscillatory behaviour of solutions acting as an inherent feature driven by complex exponential bundles of solutions. Firstly, a shooting method approach is used to show the existence of such instabilities. Afterwards, the existence of solutions is assessed in the self-similar approach and characterized by analytical and numerical evidences for a single equation in and for the complete system. Finally, a positive kernel is shown to exist and a sharp estimation is obtained.Uniqueness of solutions of the first and second initial-boundary value problems for parabolic systems in bounded domains on the planehttps://zbmath.org/1472.352162021-11-25T18:46:10.358925Z"Baderko, E. A."https://zbmath.org/authors/?q=ai:baderko.elena-a"Cherepova, M. F."https://zbmath.org/authors/?q=ai:cherepova.m-fSummary: The first and second initial-boundary value problems are considered for a second-order Petrovskii parabolic system with variable coefficients in a bounded domain with nonsmooth lateral boundaries on the plane. The uniqueness of solution of these problems is proved in the class of functions that are continuous together with the first spatial derivative in the closure of the domain.Well posedness of general cross-diffusion systemshttps://zbmath.org/1472.352172021-11-25T18:46:10.358925Z"Choquet, Catherine"https://zbmath.org/authors/?q=ai:choquet.catherine"Rosier, Carole"https://zbmath.org/authors/?q=ai:rosier.carole"Rosier, Lionel"https://zbmath.org/authors/?q=ai:rosier.lionelSummary: The paper is devoted to the mathematical analysis of the Cauchy problem for general cross-diffusion systems without any assumption about its entropic structure. A global existence result of nonnegative solutions is obtained by applying a classical Schauder fixed point theorem. The proof is upgraded for enhancing the regularity of the solution, namely its gradient belongs to the space \(L^r((0, T) \times {\Omega})\) for some \(r > 2\). To this aim, the Schauder's strategy is coupled with an extension of Meyers regularity result for linear parabolic equations. We show how this approach allows to prove the well-posedness of the problem using only assumptions prescribing and admissibility range for the \textit{ratios} between the diffusion and cross-diffusion coefficients. The results are compared with those that are reachable with an additional regularity assumption on the parabolic operator, namely a small BMO assumption for its coefficients. Finally, the question of the maximal principle is also addressed, especially when source terms are incorporated in the equation in order to ensure the confinement of the solution.A quasilinear predator-prey model with indirect prey-taxishttps://zbmath.org/1472.352182021-11-25T18:46:10.358925Z"Xing, Jie"https://zbmath.org/authors/?q=ai:xing.jie"Zheng, Pan"https://zbmath.org/authors/?q=ai:zheng.pan"Pan, Xu"https://zbmath.org/authors/?q=ai:pan.xuSummary: This paper deals with a quasilinear predator-prey model with indirect prey-taxis
\[
\begin{cases}
u_t=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla w)+rug(v)-uh(u), & (x,t)\in \Omega \times (0,\infty ),\\
w_t=d_w\Delta w- \mu w+\alpha v, & (x,t)\in \Omega \times (0,\infty ),\\
v_t=d_v\Delta v+f(v)-ug(v), & (x,t)\in \Omega \times (0,\infty ),
\end{cases}
\]
under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb{R}^n\), \(n\ge 1\), where \(d_w, d_v, \alpha, \mu, r>0\) and the functions \(g,h,f \in C^2([0,\infty ))\). The nonlinear diffusivity \(D\) and chemosensitivity \(S\) are supposed to satisfy
\[
D(s)\ge a(s+1)^{-\gamma} \quad \text{and} \quad 0\le S(s)\le bs(s+1)^{\beta -1} \quad\text{for all}\quad s\ge 0,
\]
with \(a,b>0\) and \(\gamma, \beta\in \mathbb{R}\). Suppose that \(\gamma+\beta <1+\frac{1}{n}\) and \(\gamma <\frac{2}{n}\), it is proved that the problem has a unique global classical solution, which is uniformly bounded in time. In addition, we derive the asymptotic behavior of globally bounded solution in this system according to the different predation conditions.Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimizationhttps://zbmath.org/1472.352192021-11-25T18:46:10.358925Z"Bramburger, Jason J."https://zbmath.org/authors/?q=ai:bramburger.jason-j"Goluskin, David"https://zbmath.org/authors/?q=ai:goluskin.davidSummary: Many monostable reaction-diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction-diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction-diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction-diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1\% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditionshttps://zbmath.org/1472.352202021-11-25T18:46:10.358925Z"Sharma, Vandana"https://zbmath.org/authors/?q=ai:sharma.vandanaSummary: We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions. Under reasonable hypotheses, we establish the existence of component wise non-negative global solutions which are uniformly bounded in the sup norm.Turing and Benjamin-Feir instability mechanisms in non-autonomous systemshttps://zbmath.org/1472.352212021-11-25T18:46:10.358925Z"van Gorder, Robert A."https://zbmath.org/authors/?q=ai:van-gorder.robert-aSummary: The Turing and Benjamin-Feir instabilities are two of the primary instability mechanisms useful for studying the transition from homogeneous states to heterogeneous spatial or spatio-temporal states in reaction-diffusion systems. We consider the case when the underlying reaction-diffusion system is non-autonomous or has a base state which varies in time, as in this case standard approaches, which rely on temporal eigenvalues, break down. We are able to establish respective criteria for the onset of each instability using comparison principles, obtaining inequalities which involve the in general time-dependent model parameters and their time derivatives. In the autonomous limit where the base state is constant in time, our results exactly recover the respective Turing and Benjamin-Feir conditions known in the literature. Our results make the Turing and Benjamin-Feir analysis amenable for a wide collection of applications, and allow one to better understand instabilities emergent due to a variety of non-autonomous mechanisms, including time-varying diffusion coefficients, time-varying reaction rates, time-dependent transitions between reaction kinetics and base states which change in time (such as heteroclinic connections between unique steady states, or limit cycles), to name a few examples.Critical periodic traveling waves for a periodic and diffusive epidemic modelhttps://zbmath.org/1472.352222021-11-25T18:46:10.358925Z"Zhang, Liang"https://zbmath.org/authors/?q=ai:zhang.liang|zhang.liang.2|zhang.liang.1|zhang.liang.3"Wang, Shuang-Ming"https://zbmath.org/authors/?q=ai:wang.shuangmingSummary: This work is devoted to the existence of time periodic traveling wave solutions with critical speed \(c^\ast\) for a seasonal and diffusive epidemic model. We adopt the approach of super- and sub-solutions and Schauder's fixed point theorem to the truncated problem combined with limiting arguments to obtain the existence of critical periodic traveling waves. This would be appropriate for the modification versions of current model, where standard incidence is replaced by other nonlinear incidence.On the crest factor for dissipative partial differential equationshttps://zbmath.org/1472.352232021-11-25T18:46:10.358925Z"Bartuccelli, Michele V."https://zbmath.org/authors/?q=ai:bartuccelli.michele-vSummary: In this work, we have introduced and then computed the so-called \textit{crest factor} associated with solutions of dissipative partial differential equations (PDEs). By taking two paradigmatic dissipative PDEs, we estimated in an explicit and accurate manner the values of the crest factor of their solutions. We then analysed and compared the estimates as a function of the positive parameter which appears in the PDEs in space dimensions one and two. These estimates shed some light on the dynamics of the fluctuations of the solutions of the two model PDEs, and therefore provide a criterion for discerning between small and large potential excursions in space for the solution of any dissipative PDE. Being able to detect between small and large intermittent fluctuations is one of the hallmarks of turbulence. We believe that the crest factor is an appropriate tool for extracting space fluctuation features in solutions of dissipative PDEs.Correction to: ``Control and controllability of PDEs with hysteresis''https://zbmath.org/1472.352242021-11-25T18:46:10.358925Z"Gavioli, Chiara"https://zbmath.org/authors/?q=ai:gavioli.chiara"Krejčí, Pavel"https://zbmath.org/authors/?q=ai:krejci.pavelCorrection of Equation 4.17 in the authors' paper [ibid. 84, No. 1, 829--847 (2021; Zbl 1470.35197)].Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg-Landau equationhttps://zbmath.org/1472.352252021-11-25T18:46:10.358925Z"Kostianko, Anna"https://zbmath.org/authors/?q=ai:kostianko.annaSummary: We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mané projectors) for semilinear parabolic systems with cross diffusion terms and illustrate it on the model example of three-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. The method combines the so-called spatial-averaging principle invented by Sell and Mallet-Paret with temporal averaging of rapid oscillations which come from cross-diffusion terms.Global existence and decay estimates for the heat equation with exponential nonlinearityhttps://zbmath.org/1472.352262021-11-25T18:46:10.358925Z"Majdoub, Mohamed"https://zbmath.org/authors/?q=ai:majdoub.mohamed"Tayachi, Slim"https://zbmath.org/authors/?q=ai:tayachi.slimSummary: In this paper, we consider the initial value problem in some Orlicz spaces for the heat equation with a nonlinearity having an exponential growth at infinity and vanishing at zero. Under some smallness conditions on the initial data and appropriate behavior near the origin for the nonlinearity, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time. We show in particular that the decay depends on the behavior of the nonlinearity near the origin.Existence of solution for a class of heat equation with double criticalityhttps://zbmath.org/1472.352272021-11-25T18:46:10.358925Z"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Boudjeriou, Tahir"https://zbmath.org/authors/?q=ai:boudjeriou.tahirSummary: In this paper, we study the following class of quasilinear heat equations
\[
\begin{cases}
u_t-\Delta_\Phi u=f(x,u) & \text{ in }\Omega,\quad t>0, \\
u=0 & \text{ on }\partial\Omega,\quad t>0, \\
u(x,0)=u_0(x) & \text{ in }\Omega,
\end{cases}
\]
where \(\Delta_\Phi u=\operatorname{div}(\varphi (x,|\nabla\varphi|)\nabla\varphi)\) and \(\Phi(x,s)=\int_0^{|s|}\varphi(x,\sigma)\sigma d\sigma\) is a generalized N-function. We suppose that \(\Omega\subset\mathbb{R}^N\) \((N\geq 2)\) is a smooth bounded domain that contains two open regions \(\Omega_N\) and \(\Omega_p\) with \(\overline{\Omega}_N\cap\overline{\Omega}_p=\emptyset\). Under some appropriate conditions, the global existence will be done by combining the Galerkin approximations with the potential well theory. Moreover, the large-time behavior of the global weak solution is analyzed. The main feature of this paper consists that \(-\Delta_\Phi u\) behaves like \(-\Delta_Nu\) on \(\Omega_N\) and \(-\Delta_pu\) on \(\Omega_p\), while the continuous function \(f:\Omega\times\mathbb{R}\to\mathbb{R}\) behaves like \(e^{\alpha |s|^{\frac{N}{N-1}}}\) on \(\Omega_N\) and \(|s|^{p_\ast-2}s\) on \(\Omega_p\) as \(|s|\to\infty\).A direct approach to quasilinear parabolic equations on unbounded domains by Brézis's theory for subdifferential operatorshttps://zbmath.org/1472.352282021-11-25T18:46:10.358925Z"Kurima, Shunsuke"https://zbmath.org/authors/?q=ai:kurima.shunsuke"Yokota, Tomomi"https://zbmath.org/authors/?q=ai:yokota.tomomiSummary: This paper deals with nonlinear diffusion equations and their approximate equations under homogeneous Neumann boundary conditions in unbounded domains with smooth bounded boundary. \textit{P. Colli} and \textit{T. Fukao} [J. Differ. Equations 260, No. 9, 6930--6959 (2016; Zbl 1334.35154)] studied similar equations in bounded domains by applying an abstract theory for doubly nonlinear evolution inclusions; however, the proof is based on compactness methods and hence the case of unbounded domains is excluded from the framework. The present paper asserts that one can solve the original problem and the approximate problem individually and directly in unbounded domains by applying Brézis theory.A duality between scattering poles and transmission eigenvalues in scattering theoryhttps://zbmath.org/1472.352292021-11-25T18:46:10.358925Z"Cakoni, Fioralba"https://zbmath.org/authors/?q=ai:cakoni.fioralba"Colton, David"https://zbmath.org/authors/?q=ai:colton.david-l"Haddar, Houssem"https://zbmath.org/authors/?q=ai:haddar.houssemSummary: In this paper, we develop a conceptually unified approach for characterizing and determining scattering poles and interior eigenvalues for a given scattering problem. Our approach explores a duality stemming from interchanging the roles of incident and scattered fields in our analysis. Both sets are related to the kernel of the relative scattering operator mapping incident fields to scattered fields, corresponding to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Our discussion includes the scattering problem for a Dirichlet obstacle where duality is between scattering poles and Dirichlet eigenvalues, and the inhomogeneous scattering problem where the duality is between scattering poles and transmission eigenvalues. Our new characterization of the scattering poles suggests a numerical method for their computation in terms of scattering data for the corresponding interior scattering problem.Parabolic-elliptic system modeling biological ion channelshttps://zbmath.org/1472.352302021-11-25T18:46:10.358925Z"Sapa, Lucjan"https://zbmath.org/authors/?q=ai:sapa.lucjanA system of parabolic equations describing transport of biological ions in one-dimensional channels, coupled through an elliptic equation for the potential, is considered. This system of Nernst-Planck-Poisson type is supplemented with biologically relevant boundary conditions of Chang-Jaffé type (the normal component of the flux is proportional to the weighted difference of concentrations across the boundary). The existence, regularity and positivity properties of local in time solutions are established.Strongly degenerate parabolic equations with variable coefficientshttps://zbmath.org/1472.352312021-11-25T18:46:10.358925Z"Watanabe, Hiroshi"https://zbmath.org/authors/?q=ai:watanabe.hiroshi.1|watanabe.hiroshi.6|watanabe.hiroshi.5|watanabe.hiroshi.2|watanabe.hiroshi|watanabe.hiroshi.4|watanabe.hiroshi.3Summary: We consider the initial value problem for a degenerate parabolic equation of the form:
\[
\partial_tu+\nabla\cdot A(x,t,u)+B(x,t,u)= \Delta\beta(x,t,u).\tag{DP}
\]
This equation has both properties of hyperbolic equations and those of parabolic equations. Thus, this is also called hyperbolic-parabolic equation. Since (DP) has nonlinear convection and diffusion terms, this describes various nonlinear convective diffusion phenomena such as filtration problems, Stefan problems and so on.\par One of the features of this article is that the equation (DP) has variable coefficients. In this case, it is difficult to prove a BV-estimate for approximate solutions to (DP). We overcome this difficulty to use the assumption which was derived by \textit{Z. Wu} and \textit{J. Zhao} [Chin. Ann. Math., Ser. B 4, 319--328 (1983; Zbl 0516.35045)]. To prove the uniqueness of generalized solutions to (DP), we refer to the method of \textit{G.-Q. Chen} and \textit{K. H. Karlsen} [Commun. Pure Appl. Anal. 4, No. 2, 241--266 (2005; Zbl 1084.35034)]. Using the arguments, we prove the existence and uniqueness of entropy solutions in the space BV to the problem.An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusionhttps://zbmath.org/1472.352322021-11-25T18:46:10.358925Z"Zheng, Jiashan"https://zbmath.org/authors/?q=ai:zheng.jiashanSummary: This paper investigates the following Keller-Segel-Stokes system with nonlinear diffusion
\[
(KSF)
\begin{cases}
n_t+u\cdot\nabla n = \Delta n^m - \nabla\cdot(n\nabla c),\quad x\in\Omega, t > 0, \\
c_t+u\cdot\nabla c = \Delta c-c+n,\quad x\in\Omega, t > 0, \\
u_t+\nabla P = \Delta u+n\nabla\phi,\quad x\in\Omega, t > 0, \\
\nabla\cdot u = 0,\quad x\in\Omega, t > 0
\end{cases}
\]
under homogeneous boundary conditions of Neumann type for \(n\) and \(c\), and of Dirichlet type for \(u\) in a three-dimensional bounded domains \(\Omega\subseteq\mathbb{R}^3\) with smooth boundary, where \(\phi\in W^{2,\infty}(\Omega),m > 0\). It is proved that if \(m > \frac{4}{3}\), then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system \(KSF\), which in view of the known results for the fluid-free system mentioned below (see Introduction) is an \textbf{optimal} restriction on \(m\).Classification of initial energy in a pseudo-parabolic equation with variable exponentshttps://zbmath.org/1472.352332021-11-25T18:46:10.358925Z"Li, Fengjie"https://zbmath.org/authors/?q=ai:li.fengjie"Liu, Jiaqi"https://zbmath.org/authors/?q=ai:liu.jiaqi"Liu, Bingchen"https://zbmath.org/authors/?q=ai:liu.bingchenSummary: This paper deals with a pseudo-parabolic equation with variable exponents, subject to homogeneous boundary conditions with initial data in \(H_0^1(\varOmega)\). By using energy functional and Nehari functional, we classify blow-up and global existence of weak solutions in variable Sobolev spaces completely for subcritical, critical, and super critical initial energy, respectively.Anisotropic \(p\)-Laplacian evolution of fast diffusion typehttps://zbmath.org/1472.352342021-11-25T18:46:10.358925Z"Feo, Filomena"https://zbmath.org/authors/?q=ai:feo.filomena"Vázquez, Juan Luis"https://zbmath.org/authors/?q=ai:vazquez.juan-luis"Volzone, Bruno"https://zbmath.org/authors/?q=ai:volzone.brunoSummary: We study an anisotropic, possibly non-homogeneous version of the evolution \(p\)-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive sharp \(L^1\)-\(L^{\infty}\) estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not common in the related literature. It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.Global existence and finite time blow-up of solutions to a nonlocal \(p\)-Laplace equationhttps://zbmath.org/1472.352352021-11-25T18:46:10.358925Z"Li, Jian"https://zbmath.org/authors/?q=ai:li.jian.2|li.jian.3|li.jian.1"Han, Yuzhu"https://zbmath.org/authors/?q=ai:han.yuzhuSummary: In this paper a class of nonlocal diffusion equations associated with a \(p\)-Laplace operator, usually referred to as \(p\)-Kirchhoff equations, are studied. By applying Galerkin's approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the \(L^2\) norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [\textit{Y. Han} and \textit{Q. Li}, Comput. Math. Appl. 75, No. 9, 3283--3297 (2018; Zbl 1409.35143)].Asymptotic behavior for curvature flow with driving force when curvature blowing uphttps://zbmath.org/1472.352362021-11-25T18:46:10.358925Z"Zhang, Longjie"https://zbmath.org/authors/?q=ai:zhang.longjieSummary: In this paper, we consider the asymptotic behavior of an axisymmetric closed curve evolving by its curvature with driving force. When the curve shrinks to a point, the asymptotic behavior will be a circle. As an easy corollary, this curve will become convex eventually. The main method in this research is the comparison principle for the ratio between extrinsic and intrinsic distance.Oscillations of the string with singuliaritieshttps://zbmath.org/1472.352372021-11-25T18:46:10.358925Z"Kamenskii, Mikhail"https://zbmath.org/authors/?q=ai:kamenskii.mikhail-igorevich"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfeng"Zvereva, Margarita"https://zbmath.org/authors/?q=ai:zvereva.margarita-borisovnaSummary: In the present paper we consider the initial boundary value problem describing discontinuous Stieltjes string oscillations. The possibility of Fourier series expansion of the solution is investigated. The analysis is based on a refined Stieltjes integral.Study of a viscoelastic wave equation with a strong damping and variable exponentshttps://zbmath.org/1472.352382021-11-25T18:46:10.358925Z"Liao, Menglan"https://zbmath.org/authors/?q=ai:liao.menglanSummary: The goal of the present paper is to study the viscoelastic wave equation with variable exponents
\[
u_{tt}-\Delta_{p(x)}u-\Delta u+\int_0^tg(t-s)\Delta u(s)\mathrm{d}s-\Delta u_t=|u|^{q(x)-2}u
\]
under initial-boundary value conditions, where the exponents of nonlinearity \(p(x)\) and \(q(x)\) are given functions. To be more precise, blow-up in finite time is proved, upper and lower bounds of the blow-up time are obtained as well. The global existence of weak solutions is presented, moreover, a general stability of solutions is obtained. This work generalizes and improves earlier results in the literature.Global existence and general decay of solutions for a quasilinear system with degenerate damping termshttps://zbmath.org/1472.352392021-11-25T18:46:10.358925Z"Ekinci, Fatma"https://zbmath.org/authors/?q=ai:ekinci.fatma"Pișkin, Erhan"https://zbmath.org/authors/?q=ai:piskin.erhan"Boulaaras, Salah Mahmoud"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Mekawy, Ibrahim"https://zbmath.org/authors/?q=ai:mekawy.ibrahimSummary: In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping, dispersion, and source terms under Dirichlet boundary condition. Under some restrictions on the initial datum and standard conditions on relaxation functions, we study global existence and general decay of solutions. The results obtained here are generalization of the previous recent work.The blow-up rate for a non-scaling invariant semilinear wave equations in higher dimensionshttps://zbmath.org/1472.352402021-11-25T18:46:10.358925Z"Hamza, Mohamed Ali"https://zbmath.org/authors/?q=ai:hamza.mohamed-ali"Zaag, Hatem"https://zbmath.org/authors/?q=ai:zaag.hatemSummary: We consider the semilinear wave equation
\[
\partial_t^2u-\Delta u=f(u),\quad (x,t)\in\mathbb{R}^N\times [0,T),\tag{1}
\]
with \(f(u)=|u|^{p-1}u\log^a(2+u^2)\), where \(p>1\) and \(a\in\mathbb{R}\), with subconformal power nonlinearity. We will show that the blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), The result in one space dimension, has been proved in [the authors, J. Math. Anal. Appl. 483, No. 2, Article ID 123652, 34 p. (2020; Zbl 1439.35088)]. Our goal here is to extend this result to higher dimensions.Logarithmic viscoelastic wave equation in three-dimensional spacehttps://zbmath.org/1472.352412021-11-25T18:46:10.358925Z"Ye, Yaojun"https://zbmath.org/authors/?q=ai:ye.yaojunSummary: The initial-boundary value problem of logarithmic viscoelastic wave equation in three space dimensions is studied. The existence of local and global solutions for this problem are proved by means of the contraction mapping criterion and potential well method. Meanwhile, the blow-up of solutions in the unstable set is also obtained.Stability of quasilinear waves in \(1+1\) dimension under null conditionhttps://zbmath.org/1472.352422021-11-25T18:46:10.358925Z"Dong, Shijie"https://zbmath.org/authors/?q=ai:dong.shijieSummary: We are interested in the global-in-time existence results for quasilinear wave equations satisfying null condition in one dimension. The main difficulty is that waves do not decay in one dimension space, and we rely on the new type of weighted energy estimates introduced in \textit{G. K. Luli} et al. [Adv. Math. 329, 174--188 (2018; Zbl 1392.35190)], where the authors proved that small data lead to global-in-time solutions for semilinear wave equations with null nonlinearities in dimension one.Lower bound of the lifespan of the solution to systems of quasi-linear wave equations with multiple propagation speedshttps://zbmath.org/1472.352432021-11-25T18:46:10.358925Z"Hoshiga, Akira"https://zbmath.org/authors/?q=ai:hoshiga.akiraSummary: We consider the Cauchy problem of systems of quasilinear wave equations in 2-dimensional space. We assume that the propagation speeds are distinct and that the nonlinearities contain quadratic and cubic terms of the first and second order derivatives of the solution. We know that if the all quadratic and cubic terms of nonlinearities satisfy Strong Null-condition, then there exists a global solution for sufficiently small initial data. In this paper, we study about the lifespan of the smooth solution, when the cubic terms in the quasi-linear nonlinearities do not satisfy the Strong null-condition. In the proof of our claim, we use the ghost weight energy method and the \(L^\infty-L^\infty\) estimates of the solution, which is slightly improved.Asymptotic behavior of global solutions to one-dimension quasilinear wave equationshttps://zbmath.org/1472.352442021-11-25T18:46:10.358925Z"Li, Mengni"https://zbmath.org/authors/?q=ai:li.mengniIn this paper, the author considered the Cauchy problem of one-dimension systems of quasilinear wave equations with null conditions. In the small data setting, based on the global existence results in [\textit{G. K. Luli} et al., Adv. Math. 329, 174--188 (2018; Zbl 1392.35190)] in the semilinear case, and [\textit{D. Zha}, Calc. Var. Partial Differ. Equ. 59, No. 3, Paper No. 94, 19 p. (2020; Zbl 1441.35176)] in the quasilinear case, she gave some further asymptotic behavior results for the global solution. First, she showed that the global solution is asymptotically free in the weighted energy sense. This result generalized the corresponding one in [loc. cit.], which gave the global solution is asymptotically free in the unweighted energy sense. Then, she showed the rigidity of the scattering field.Initial boundary value problem for fractional \(p \)-Laplacian Kirchhoff type equations with logarithmic nonlinearityhttps://zbmath.org/1472.352452021-11-25T18:46:10.358925Z"Shi, Peng"https://zbmath.org/authors/?q=ai:shi.peng.1|shi.peng"Jiang, Min"https://zbmath.org/authors/?q=ai:jiang.min"Zeng, Fugeng"https://zbmath.org/authors/?q=ai:zeng.fugeng"Huang, Yao"https://zbmath.org/authors/?q=ai:huang.yaoSummary: In this paper, we study the initial boundary value problem for a class of fractional \(p \)-Laplacian Kirchhoff type diffusion equations with logarithmic nonlinearity. Under suitable assumptions, we obtain the extinction property and accurate decay estimates of solutions by virtue of the logarithmic Sobolev inequality. Moreover, we discuss the blow-up property and global boundedness of solutions.Invariance solutions and blow-up property for edge degenerate pseudo-hyperbolic equations in edge Sobolev spaceshttps://zbmath.org/1472.352462021-11-25T18:46:10.358925Z"Cattani, Carlo"https://zbmath.org/authors/?q=ai:cattani.carlo"Kalleji, Morteza Koozehgar"https://zbmath.org/authors/?q=ai:kalleji.morteza-koozehgarSummary: This article is dedicated to study of the initial-boundary value problem of edge pseudo-hyperbolic system with damping term on the manifold with edge singularity. First, we will discuss about the invariance of solution set of a class of edge degenerate pseudo-hyperbolic equations on the edge Sobolev spaces. Then, by using a family of modified potential wells and concavity methods, it is obtained existence and nonexistence results of global solutions with exponential decay and is shown the blow-up in finite time of solutions on the manifold with edge singularities.
For the entire collection see [Zbl 1471.34005].Correction to: ``Two generalized Tricomi equations''https://zbmath.org/1472.352472021-11-25T18:46:10.358925Z"Paronetto, Fabio"https://zbmath.org/authors/?q=ai:paronetto.fabioFigure 1 caption and some lines in Sections 4.2 and 4.3 in the author's paper [ibid. 60, No. 3, Paper No. 84, 28 p. (2021; Zbl 1462.35206)] are corrected.Nonlocal Dezin problem for a mixed type equation of the second kindhttps://zbmath.org/1472.352482021-11-25T18:46:10.358925Z"Khairullin, R. S."https://zbmath.org/authors/?q=ai:khairullin.r-sSummary: For the equation \(u_{{xx}}+yu_{{yy}}+\alpha u_y=0\) with parameter \(\alpha \leq -1/2\) given in the mixed domain that is the rectangle \([0,1]\times [-\beta ,\gamma ]\), where \(\beta >0\) and \(\gamma >0 \), we investigate Dezin's problem in which a periodicity condition is set on the vertical rectangle sides, the values of the unknown function are specified on the top side, matching conditions are given on the singular line, and a nonlocal condition relating the values of the unknown function on the singular line to the values of the normal derivative of this function is specified on the bottom side. The solution of the problem is constructed in the form of a series. Sufficient conditions ensuring the existence of a solution are found for the given functions and the parameters \(\beta\) and \(\gamma \). A uniqueness criterion is established.Initial-boundary value problem for a three-dimensional equation of the parabolic-hyperbolic typehttps://zbmath.org/1472.352492021-11-25T18:46:10.358925Z"Sabitov, K. B."https://zbmath.org/authors/?q=ai:sabitov.kamil-basirovich"Sidorov, S. N."https://zbmath.org/authors/?q=ai:sidorov.stanislav-nikolaevichSummary: We study an initial-boundary value problem for an equation of the mixed parabolic-hyperbolic type in a rectangular parallelepiped. A uniqueness criterion is established. The solution is constructed in the form of a series in an orthogonal function system. The problem of small denominators depending on two positive integer arguments arises when justifying the convergence of this series. Sufficient conditions are found for the uniform separation of the denominators from zero; this permits us to prove the convergence of the series in the class of regular solutions and the stability of the solution with respect to perturbations of the boundary functions.On Courant's nodal domain property for linear combinations of eigenfunctions. IIhttps://zbmath.org/1472.352502021-11-25T18:46:10.358925Z"Bérard, Pierre"https://zbmath.org/authors/?q=ai:berard.pierre-h"Helffer, Bernard"https://zbmath.org/authors/?q=ai:helffer.bernardSummary: Generalizing Courant's nodal domain theorem, the ``Extended Courant property'' is the statement that a linear combination of the first neigenfunctions has at most nnodal domains. In a previous paper [Doc. Math. 23, 1561--1585 (2018; Zbl 1403.35192)], we gave simple counterexamples to this property, including convex domains. In the present paper, using some input from numerical computations, we pursue the investigation of the Extended Courant property with two new examples, the equilateral rhombus and the regular hexagon.
For the entire collection see [Zbl 1468.35004].Estimates of Dirichlet eigenvalues for a class of sub-elliptic operatorshttps://zbmath.org/1472.352512021-11-25T18:46:10.358925Z"Chen, Hua"https://zbmath.org/authors/?q=ai:chen.hua"Chen, Hong-Ge"https://zbmath.org/authors/?q=ai:chen.honggeSummary: Let \(\Omega\) be a bounded connected open subset in \(\mathbb{R}^n\) with smooth boundary \(\partial\Omega\). Suppose that we have a system of real smooth vector fields \(X = (X_1,X_2, \ldots , X_m )\) defined on a neighborhood of \(\overline{\Omega}\) that satisfies the Hörmander's condition. Suppose further that \(\partial\Omega\) is non-characteristic with respect to \(X\). For a self-adjoint sub-elliptic operator \(\Delta_X = - \sum_{i = 1}^m X_i^\ast X_i\) on \(\Omega\), we denote its \(k\mathrm{th}\) Dirichlet eigenvalue by \(\lambda_k\). We will provide a uniform upper bound for the sub-elliptic Dirichlet heat kernel. We will also give an explicit sharp lower bound estimate for \(\lambda_k\), which has a polynomially growth in \(k\) of the order related to the generalized Métivier index. We will establish an explicit asymptotic formula of \(\lambda_k\) that generalizes the Métivier's results in 1976. Our asymptotic formula shows that under a certain condition, our lower bound estimate for \(\lambda_k\) is optimal in terms of the growth of \(k\). Moreover, the upper bound estimate of the Dirichlet eigenvalues for general sub-elliptic operators will also be given, which, in a certain sense, has the optimal growth order.Upper bounds for Steklov eigenvalues of submanifolds In Euclidean space via the intersection indexhttps://zbmath.org/1472.352522021-11-25T18:46:10.358925Z"Colbois, Bruno"https://zbmath.org/authors/?q=ai:colbois.bruno"Gittins, Katie"https://zbmath.org/authors/?q=ai:gittins.katieSummary: We obtain upper bounds for the Steklov eigenvalues \(\sigma_k(M)\) of a smooth, compact, \(n\)-dimensional submanifold \(M\) of Euclidean space with boundary \(\Sigma\) that involve the intersection indices of \(M\) and of \(\Sigma\). One of our main results is an explicit upper bound in terms of the intersection index of \(\Sigma\), the volume of \(\Sigma\) and the volume of \(M\) as well as dimensional constants. By also taking the injectivity radius of \(\Sigma\) into account, we obtain an upper bound that has the optimal exponent of \(k\) with respect to the asymptotics of the Steklov eigenvalues as \(k\to\infty\).Widths of resonances above an energy-level crossinghttps://zbmath.org/1472.352532021-11-25T18:46:10.358925Z"Fujiié, S."https://zbmath.org/authors/?q=ai:fujiie.setsuro"Martinez, A."https://zbmath.org/authors/?q=ai:martinez.andre"Watanabe, T."https://zbmath.org/authors/?q=ai:watanabe.takuyaThe authors study a \(2\times2\) Schrödinger operator
\[
Pu = Eu,\qquad P = \left( \begin{matrix} P_1 & hW\\
hW^* & P_2 \end{matrix} \right),
\]
where \(D_x:=-i\frac{d}{d x}\), \(P_j:=h^2D_x^2 + V_j(x)\) (\(j=1,2\)), \(W=W(x,hD_x)\) is a semiclassical differential operator, and \(W^*\) is the formal adjoint of \(W\). The main aim is to study the asymptotic distribution of resonances in the semiclassical limit \(h\to 0_+\) in a neighborhood of a fixed real energy \(E_0\).
A series of conditions is supposed:
\textbf{Assumption (A1)} \(V_1(x)\), \(V_2(x)\) are real-valued analytic functions on \(\mathbb{R}\), and extend to holomorphic functions in the complex domain, \(\mathcal{S}=\{x\in\mathbb{C}\,;\,|\mathrm{IM}\,\, x|<\delta_0\langle\mathrm{Re}\, \,x\rangle\}\), where \(\delta_0>0\) is a constant, and \(\langle t\rangle:=(1+|t|^2)^{1/2}\).
\textbf{Assumption (A2)} For \(j=1,2\), \(V_j\) admits limits as \(\mathrm{Re}\,\, x\to \pm\infty\) in \(\mathcal{S}\), and they satisfy
\[
\begin{aligned}
\lim_{\substack{\mathrm{Re}\,\,x\to -\infty \\ x\in \mathcal{S}}} V_1(x)>E_0\, ;\, \lim_{\substack{\mathrm{Re}\,\,x\to -\infty \\ x\in \mathcal{S}}} V_2(x)>E_0\, ;\\
\lim_{\substack{\mathrm{Re}\,\,x\to +\infty \\ x\in \mathcal{S}}} V_1(x)>E_0\, ;\, \lim_{\substack{\mathrm{Re}\,\,x\to +\infty \\ x\in \mathcal{S}}} V_2(x)<E_0.
\end{aligned}
\]
\textbf{Assumption (A3)} There exist three numbers \(a<b<0<c\) such that
\[
V_1(a)=V_1(c)=V_2(b)=E_0, V_1'(a) < 0, V_1'(c) > 0, V_2'(b) < 0,
\]
and that
\[
\begin{array}{ll}
V_1>E_0\text{ on }(-\infty, a)\cup (c,+\infty), V_1<E_0\text{ on }(a,c), V_2>E_0\text{ on }(-\infty, b), V_2<E_0\text{ on }(b,+\infty).
\end{array}
\]
\textbf{Assumption (A4)} The set \(\{x\in \mathbb{R}; V_1(x)=V_2(x)\, , \, V_1(x)\leq E_0\, ,\, V_2(x)\leq E_0\}\) is reduced to \(\{0\}\), and one has \(V_1(0)=V_2(0)=0\), \(V'_1(0)>0\), \(V_2'(0)<0\).
In particular, in the phase-space, the characteristic sets \(\Gamma_j:=\{\xi^2 +V_j(x)=E_0\}\) (\(j=1,2\)) intersect transversally at \((0,\pm\sqrt{E_0})\).
\textbf{Assumption (A5)} \(W(x,hD_x)\) is a first order differential operator,
\[
W(x,hD_x)=r_0(x)+ir_1(x)hD_x,
\]
where \(r_0\) and \(r_1\) are bounded analytic functions on \(\mathcal{S}\), are real on the real, and such that \(W\) is elliptic at the crossing points \((0,\pm\sqrt{E_0})\), that is, \( (r_0(0),r_1(0)) \not=(0,0). \)
Under the above assumptions, in a neighbourhood of the energy \(E_0\), the spectrum of \(P\) is essential only. Let \(\mathrm{Res}(P)\) be the set of the resonances of the operator \(P\). For \(E\in \mathbb{C}\) close enough to \(E_0\), the authors define the action,
\[
\mathcal{A}(E):= \int_{a(E)}^{c(E)}\sqrt{ E-V_1(t)} \, dt,
\]
where \(a(E)\) (respectively \(c(E)\)) is the unique solution of \(V_1(x)=E\) close to \(a\) (respectively close to \(c\)). In this situation, \(\mathcal{A}(E)\) is an analytic function of \(E\) near \(E_0\) and \(\mathcal{A}'(E)\) is strictly positive for any real \(E\) near \(E_0\). Then fix \(\delta_0>0\) sufficiently small and \(C_0>0\) arbitrarily large and let
\[
\mathcal{D}_h(\delta_0,C_0):= [E_0-\delta_0, E_0+\delta_0]-i[0,C_0h].
\]
For \(h>0\) and \(k\in\mathbb{Z}\) such that \((k+\frac12)\pi h\) belongs to \(\mathcal{A}( [E_0-2\delta_0, E_0+2\delta_0])\), let
\[
e_k(h):=\mathcal{A}^{-1}\left( (k+\frac12)\pi h\right).
\]
The main result is as follows. Under Assumptions (A1)-(A5), there exists \(\delta_0>0\) such that for any \(C_0>0\), one has, for \(h>0\) small enough
\[
\mathrm{Res}\,(P)\cap \mathcal{D}_h(\delta_0, C_0) =\{E_k(h); k\in\mathbb{Z}\}\cap\mathcal{D}_h(\delta_0, C_0)
\]
where the \(E_k(h)\)'s are complex numbers that satisfy \begin{align*}
\mathrm{Re}\, \,E_k(h) &= e_k(h) + \mathcal{O}(h^{2}),\\
\mathrm{Im}\, \,E_k(h) &= - C(e_k(h))h^2+ \mathcal{O}(h^{7/3}), \end{align*}
uniformly as \(h \to 0\). Here
\[
C(E)=\frac {\pi }{\gamma\mathcal{A}'(E)} \left |r_0(0)E^{-\frac 14}\sin \left(\frac{\mathcal{B}(E)}{h} + \frac{\pi}{4}\right) + r_1(0)E^{\frac 14}\cos \left(\frac{\mathcal{B}(E)}{h} + \frac{\pi}{4}\right)\right |^2
\]
with \(\gamma := V_1'(0)-V_2'(0) > 0\) and
\[
\mathcal{B}(E):=\int_{b(E)}^0\!\!\!\!\sqrt{E-V_2(x)}dx+\int_0^{c(E)}\!\!\!\!\!\sqrt{E-V_1(x)}dx,
\]
where \(b(E)\) is the unique root of \(V_2(x)=E\) close to \(b\).Spectral stability estimates of Dirichlet divergence form elliptic operatorshttps://zbmath.org/1472.352542021-11-25T18:46:10.358925Z"Gol'dshtein, Vladimir"https://zbmath.org/authors/?q=ai:goldshtein.vladimir"Pchelintsev, Valerii"https://zbmath.org/authors/?q=ai:pchelintsev.valerii"Ukhlov, Alexander"https://zbmath.org/authors/?q=ai:ukhlov.alexanderThe paper is aimed on applying quasiconformal mappings to spectral stability estimates of the Dirichlet eigenvalues of \(A\)-divergent form elliptic operators
\[
L_{A}=-\text{div} [A(w) \nabla g(w)]\in \widetilde{\Omega}, \quad w|_{\partial \widetilde{\Omega}}=0,
\]
in non-Lipschitz domains \(\widetilde{\Omega} \subset \mathbb{C}\) with \(2 \times 2\) symmetric matrix functions \(A(w)=\left\{a_{kl}(w)\right\}\), \(\textrm{det} A=1\), with measurable entries satisfying the uniform ellipticity condition.
The main results of the article concern to spectral stability estimates in domains that the authors call as \(A\)-quasiconformal \(\beta\)-regularity domains. Namely, a simply connected domain \(\widetilde{\Omega} \subset \mathbb{C}\) is called an \(A\)-quasiconformal \(\beta\)-regular domain about a simply connected domain \({\Omega} \subset \mathbb{C}\) if
\[
\iint\limits_{\widetilde{\Omega}} |J(w, \varphi)|^{1-\beta}~dudv < \infty, \,\,\,\beta>1,
\]
where \(J(w, \varphi)\) is a Jacobian of an \(A\)-quasiconformal mapping \(\varphi: \widetilde{\Omega}\to\Omega\).
The main result of the article states that, if a domain \(\widetilde{\Omega}\) is \(A\)-quasiconformal \(\beta\)-regular about \(\Omega\), then for any \(n\in \mathbb{N}\) the following spectral stability estimates hold:
\[
|\lambda_n[I, \Omega]-\lambda_n[A, \widetilde{\Omega}]| \leq c_n A^2_{\frac{4\beta}{\beta -1},2}(\Omega) \left(|\Omega|^{\frac{1}{2\beta}} + \|J_{\varphi^{-1}}\,|\,L^{\beta}(\Omega)\|^{\frac{1}{2}} \right) \cdot \|1-J_{\varphi^{-1}}^{\frac{1}{2}}\,|\,L^{2}(\Omega)\|,
\]
where \(c_n=\max\left\{\lambda_n^2[A, \Omega], \lambda_n^2[A, \widetilde{\Omega}]\right\}\), \(J_{\varphi^{-1}}\) is a Jacobian of an \(A^{-1}\)-quasiconformal mapping \(\varphi^{-1}:\Omega\to\widetilde{\Omega}\), and
\[
A_{\frac{4\beta}{\beta -1},2}(\Omega) \leq \inf\limits_{p\in \left(\frac{4\beta}{3\beta -1},2\right)} \left(\frac{p-1}{2-p}\right)^{\frac{p-1}{p}} \frac{\left(\sqrt{\pi}\cdot\sqrt[p]{2}\right)^{-1}|\Omega|^{\frac{\beta-1}{4\beta}}}{\sqrt{\Gamma(2/p) \Gamma(3-2/p)}}~~.
\]Semiclassical resonances generated by crossings of classical trajectorieshttps://zbmath.org/1472.352552021-11-25T18:46:10.358925Z"Higuchi, Kenta"https://zbmath.org/authors/?q=ai:higuchi.kentaSummary: We consider a \(2\times 2\) system of one-dimensional semiclassical Schrödinger operators with small interactions with respect to the semiclassical parameter \(h\). We study the asymptotics in the semiclassical limit of the resonances near a non-trapping energy for both corresponding classical Hamiltonians. We show the existence of resonances of width \(T^{-1}h\log (1/h)\), contrary to the scalar case, under the condition that two classical trajectories cross and compose a periodic trajectory with period \(T\).Well-posedness of Weinberger's center of mass by Euclidean energy minimizationhttps://zbmath.org/1472.352562021-11-25T18:46:10.358925Z"Laugesen, R. S."https://zbmath.org/authors/?q=ai:laugesen.richard-snyderSummary: The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure.Quasimode, eigenfunction and spectral projection bounds for Schrödinger operators on manifolds with critically singular potentialshttps://zbmath.org/1472.352572021-11-25T18:46:10.358925Z"Blair, Matthew D."https://zbmath.org/authors/?q=ai:blair.matthew-d"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannick"Sogge, Christopher D."https://zbmath.org/authors/?q=ai:sogge.christopher-dSummary: We obtain quasimode, eigenfunction and spectral projection bounds for Schrödinger operators, \(H_V=-\Delta_g+V(x)\), on compact Riemannian manifolds \((M,g)\) of dimension \(n\geq 2\), which extend the results of the third author [J. Funct. Anal. 77, No. 1, 123--138 (1988; Zbl 0641.46011)] corresponding to the case where \(V\equiv 0\). We are able to handle critically singular potentials and consequently assume that \(V\in L^{\frac{n}{2}}(M)\) and/or \(V\in\mathcal{K}(M)\) (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where \(V\equiv 0\) that go back to the third author [loc. cit.] as well as ones which arose in the work of \textit{C. E. Kenig} et al. [Duke Math. J. 55, 329--347 (1987; Zbl 0644.35012)] in the study of ``uniform Sobolev estimates'' in \(\mathbb{R}^n\). We also use techniques from more recent developments of several authors concerning variations on the latter theme in the setting of compact manifolds. Using the spectral projection bounds we can prove a number of natural \(L^p\rightarrow L^p\) spectral multiplier theorems under the assumption that \(V\in L^{\frac{n}{2}}(M)\cap\mathcal{K}(M)\). Moreover, we can also obtain natural analogs of the original \textit{R. S. Strichartz} estimates [Duke Math. J. 44, 705--714 (1977; Zbl 0372.35001)] for solutions of \((\partial_t^2-\Delta +V)u=0\). We also are able to obtain analogous results in \(\mathbb{R}^n\) and state some global problems that seem related to works on absence of embedded eigenvalues for Schrödinger operators in \(\mathbb{R}^n\) (e.g., \textit{A. D. Ionescu} and \textit{D. Jerison} [Geom. Funct. Anal. 13, No. 5, 1029--1081 (2003; Zbl 1055.35098)]; \textit{D. Jerison} and \textit{C. E. Kenig} [Ann. Math. (2) 121, 463--494 (1985; Zbl 0593.35119)]; \textit{C. E. Kenig} and \textit{N. Nadirashvili} [Math. Res. Lett. 7, No. 5--6, 625--630 (2000; Zbl 0973.35064)]; \textit{H. Koch} and \textit{D. Tataru} [J. Reine Angew. Math. 542, 133--146 (2002; Zbl 1222.35050)]; \textit{I. Rodnianski} and \textit{W. Schlag} [Invent. Math. 155, No. 3, 451--513 (2004; Zbl 1063.35035)]).Remark on justification of asymptotics of spectra of cylindrical waveguides with periodic singular perturbations of boundary and coefficientshttps://zbmath.org/1472.352582021-11-25T18:46:10.358925Z"Gómez, D."https://zbmath.org/authors/?q=ai:gomez.delfina"Nazarov, S. A."https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovich"Orive-Illera, R."https://zbmath.org/authors/?q=ai:orive.rafael"Pérez-Martínez, M.-E."https://zbmath.org/authors/?q=ai:perez-martinez.maria-eugeniaSummary: To perform an asymptotic analysis of spectra of singularly perturbed periodic waveguides, it is required to estimate remainders of asymptotic expansions of eigenvalues of a model problem on the periodicity cell uniformly with respect to the Floquet parameter. We propose two approaches to this problem. The first is based on the max-min principle and is sufficiently easily realized, but has a restricted application area. The second is more universal, but technically complex since it is required to prove the unique solvability of the problem on the cell for some value of the spectral parameter and the Floquet parameter in a nonempty closed segment, which is verified by constructing an almost inverse operator of the operator of an inhomogeneous model problem in variational setting. We consider boundary value problems on the simplest periodicity cell: a rectangle with a row of fine holes.Behavior of eigenvalues of certain Schrödinger operators in the rational Dunkl settinghttps://zbmath.org/1472.352592021-11-25T18:46:10.358925Z"Hejna, Agnieszka"https://zbmath.org/authors/?q=ai:hejna.agnieszkaSummary: For a normalized root system \(R\) in \(\mathbb{R}^N\) and a multiplicity function \(k\ge 0\) let \(\mathbf{N}=N+\sum_{\alpha \in R} k(\alpha)\). We denote by \(dw(\mathbf{x})=\prod_{\alpha \in R}|\langle\mathbf{x},\alpha \rangle |^{k(\alpha)}d\mathbf{x}\) the associated measure in \(\mathbb{R}^N\). Let \(L=-\varDelta +V, V\ge 0\), be the Dunkl-Schrödinger operator on \(\mathbb{R}^N\). Assume that there exists \(q >\max (1,\frac{\mathbf{N}}{2})\) such that \(V\) belongs to the reverse Hölder class \(\mathrm{RH}^q(dw)\). For \(\lambda>0\) we provide upper and lower estimates for the number of eigenvalues of \(L\) which are less or equal to \(\lambda\). Our main tool in the Fefferman-Phong type inequality in the rational Dunkl setting.Subharmonic functions in scattering theoryhttps://zbmath.org/1472.352602021-11-25T18:46:10.358925Z"Denisov, Sergey A."https://zbmath.org/authors/?q=ai:denisov.sergey-aSummary: We present a method that uses the properties of subharmonic functions to control spatial asymptotics of Green's kernel of multidimensional Schrödinger operator with rough potential.Analysis of two transmission eigenvalue problems with a coated boundary conditionhttps://zbmath.org/1472.352612021-11-25T18:46:10.358925Z"Harris, I."https://zbmath.org/authors/?q=ai:harris.ian-g|harris.isaac|harris.irina|harris.isadore|harris.ian-rThe author of this article considers two transmission problems arising in scattering theory for a media with a coated boundary. Specifically, the electromagnetic transmission eigenvalue problem and the acoustic zero-index transmission eigenvalue problem is investigated. Transmission problems are new eigenvalue problems and the corresponding eigenvalues can be determined from scattering data. Additionally, they contain information on material properties of the media which one wants to obtain for example in non-destructive testing. In this work, it is proven that infinitely many real eigenvalues exists and that they depend monotonically both on the refractive index and the boundary parameters. Further, it is shown that the classical eigenvalue problem arises when the boundary parameter tends to zero or to infinity. Numerical results are given in two dimensions and support the theoretical findings.First Robin eigenvalue of the \(p\)-Laplacian on Riemannian manifoldshttps://zbmath.org/1472.352622021-11-25T18:46:10.358925Z"Li, Xiaolong"https://zbmath.org/authors/?q=ai:li.xiaolong"Wang, Kui"https://zbmath.org/authors/?q=ai:wang.kuiThe purpose of this paper is to study the first Robin eigenvalue of the \(p\)-Laplacian on compact Riemannian manifolds with boundary. The authors establish Cheng's eigenvalue comparison theorem (Theorem 1.1) and sharp bounds for the first Robin eigenvalue of the \(p\)-Laplacian (Theorem 1.4).
The authors denote \((M^n, g)\) an \(n\)-dimensional smooth compact Riemannian manifold with smooth boundary \(\partial M \not= \emptyset \). Let \(\Delta_p\) denote the \(p\)-Laplacian defined for \(1 < p < \infty \) by
\[
\Delta_p u := \mbox{div}(|\nabla u|^{p-2}\nabla u)\,,
\]
for \(u \in W^{1,p}(M)\). The authors consider the following eigenvalue problem with Robin boundary condition
\[
\begin{cases}
-\Delta_p v = \lambda |v|^{p-2}v,\quad\mbox{in}\;\; M, \cr \frac{\partial v}{\partial \nu} \,|\!\nabla v|^{p-2} + \alpha|v|^{p-2} v = 0, \quad\mbox{on}\;\; \partial M,
\end{cases}\tag{1}
\]
where \(\nu\) denotes the outward unit normal vector field along \(\partial M\) and \(\alpha \in \mathbb{R} \) is called the Robin parameter. The first Robin eigenvalue for \(\Delta_p\), denoted by \(\lambda_p(M, \alpha)\), is the smallest number such that (1) admits a weak solution in the distributional sense.
Theorem 1.1. Let \(M^n(\kappa )\) denote the simply-connected \(n\)-dimensional space form with constant sectional curvature \(\kappa\) and let \(V(\kappa , R)\) be a geodesic ball of radius \(R\) in \(M^n(\kappa)\). Let \(B_R(x_0) \subset M\) be the geodesic ball of radius \(R\) centered at \(x_0\). (We always have \(R < \frac{\pi}{\sqrt{\kappa}}\) if \(\kappa > 0\) in view of the Myers theorem).
(1) Suppose \, \(\mathrm{Ric}\ge (n - 1)\kappa\) on \(B_R(x_0)\). Then \[\lambda_p(B_R(x_0), \alpha) \le \lambda_p(V(\kappa , R), \alpha),\text{ if }\alpha > 0,\]
\[\lambda_p(B_R(x_0), \alpha) \ge \lambda_p(V(\kappa , R), \alpha),\text{ if }\alpha < 0. \]
(2) Let \(\Omega \subset B_R(x_0)\) be a domain with smooth boundary. Suppose \(\mathrm{Sect} \le \kappa\) on \(\Omega\) and \(R\) is less than the injectivity radius at \(x_0\). Then \[\lambda_p(\Omega , \alpha) \ge \lambda_p(V(\kappa , R), \alpha),\text{ if }\alpha > 0,\]
\[\lambda_p(\Omega , \alpha) \le \lambda_p(V(\kappa , R), \alpha),\text{ if }\alpha < 0.\] Moreover, the equality holds if and only if \(B_R(x_0)\) (or \(\Omega\) ) is isometric to \(V(\kappa , R)\).
Now let \(R\) denote the inradius of \(M\) defined by \(R = \sup\{d(x, \partial M) : x \in M\}\). Let \(C_{\kappa ,\Lambda}(t)\) be the unique solution of \[ \begin{cases} C_{\kappa ,\Lambda}'' + \kappa\, C_{\kappa ,\Lambda}(t) = 0,\cr C_{\kappa ,\Lambda}(0) = 1, \cr C_{\kappa ,\Lambda}'(0) = -\Lambda, \end{cases} \] and define \[ T\kappa ,\Lambda(t) := \frac{C_{\kappa ,\Lambda}'(t)}{C_{\kappa ,\Lambda}(t)} . \]\\
The second main theorem states the following:
Theorem 1.4. Suppose that the Ricci curvature of \(M\) is bounded from below by \((n - 1)\kappa\) and the mean curvature of \(\partial M\) is bounded from below by \((n - 1)\Lambda\) for some \(\kappa , \Lambda \in \mathbb{R}\). Then
\[\lambda_p(M, \alpha) \ge \bar{\lambda}_p ([0, R], \alpha,\text{ if }\alpha > 0,\]
\[\lambda_p(M, \alpha) \le \bar{\lambda}_p ([0, R], \alpha),\text{ if }\alpha < 0,\]
where \(\bar{\lambda}_p ([0, R], \alpha)\) is the first eigenvalue of the one-dimensional eigenvalue problem
\[ \begin{cases}
(p - 1)|\varphi'|^{p-2}\varphi'' + (n - 1)T_{\kappa ,\Lambda}|\varphi' |^{p-2}\varphi' = -\lambda |\varphi|^{p-2}\varphi, \cr |\varphi'(0)|^{p-2}\varphi'(0) = \alpha|\varphi(0)|^{p-2}\varphi(0),\cr \varphi'(R) = 0.
\end{cases} \]
Moreover, the equality occurs if and only if \((M, g)\) is a \((\kappa , \Lambda)\)-model space defined in Definition 6.1.Boundary value problems for the Euler-Poisson-Darboux equationhttps://zbmath.org/1472.352632021-11-25T18:46:10.358925Z"Lyakhov, L. N."https://zbmath.org/authors/?q=ai:lyakhov.l-n"Yeletskikh, K. S."https://zbmath.org/authors/?q=ai:yeletskikh.konstantin-sergeevich"Sanina, E. L."https://zbmath.org/authors/?q=ai:sanina.e-lAuthors' abstract: We consider the Cauchy problem for the Euler-Poisson-Darboux equation with boundary conditions. The Bessel operators in the equation can have different parameters. We establish a representation of the solution in the form of the Poisson formula with a special shift generated by the product of cylindrical functions of the first kind and different orders.Erratum to: ``Cauchy problem and exponential stability for the inhomogeneous Landau equation''https://zbmath.org/1472.352642021-11-25T18:46:10.358925Z"Carrapatoso, Kleber"https://zbmath.org/authors/?q=ai:carrapatoso.kleber"Tristani, Isabelle"https://zbmath.org/authors/?q=ai:tristani.isabelle"Wu, Kung-Chien"https://zbmath.org/authors/?q=ai:wu.kung-chienFrom the text: We correct a mistake in our paper [ibid. 221, No. 1, 363--418 (2016; Zbl 1342.35205)].
In the study of the linearized equation in Section 2, estimate (2.27) in Lemma
2.8 is not correct, and this error is then straithgforwardly propagated to (2.5) in
Theorem 2.3 and to the second estimate in Corollary 3.1. This last estimate is then
used to treat the nonlinear equation in the proof of Proposition 3.7.
In this erratum we first show another (weaker) regularity estimate in the place
of (2.27), which is then propagated to Theorem 2.3 and Corollary 3.1. Finally we
show how the last estimate is used in the proof of Proposition 3.7.
The results of the original paper remain unchanged, and the new regularity
estimate we shall prove here is a direct consequence of the techniques already
presented in the paper. The only modification to perform is in the condition (H0)-(i) that need to be changed into \(k > 3\gamma / 2 + 7 + 3/2\).Existence of a nonequilibrium steady state for the nonlinear BGK equation on an intervalhttps://zbmath.org/1472.352652021-11-25T18:46:10.358925Z"Evans, Josephine"https://zbmath.org/authors/?q=ai:evans.josephine"Menegaki, Angeliki"https://zbmath.org/authors/?q=ai:menegaki.angelikiSummary: We show existence of a nonequilibrium steady state for the one-dimensional, nonlinear BGK model on an interval with diffusive boundary conditions. These boundary conditions represent the coupling of the system with two heat reservoirs at different temperatures. The result holds when the boundary temperatures at the two ends are away from the equilibrium case, as our analysis is not perturbative around the equilibrium. We employ a fixed point argument to reduce the study of the model with nonlinear collisional interactions to the linear BGK model.Quantitative regularity for the Navier-Stokes equations via spatial concentrationhttps://zbmath.org/1472.352662021-11-25T18:46:10.358925Z"Barker, Tobias"https://zbmath.org/authors/?q=ai:barker.tobias"Prange, Christophe"https://zbmath.org/authors/?q=ai:prange.christopheThe authors consider the Cauchy problem for the 3D incompressible Navier-Stokes equations on \(\mathbb R^3\). The aim is to get quantitative estimates for blow-up conditions. They develop a strategy for obtaining new quantitative estimates of the Navier-Stokes equations. The main novelty is that the strategy allows obtaining local quantitative estimates.Compressible Navier-Stokes equations with heterogeneous pressure lawshttps://zbmath.org/1472.352672021-11-25T18:46:10.358925Z"Bresch, Didier"https://zbmath.org/authors/?q=ai:bresch.didier"Jabin, Pierre-Emmanuel"https://zbmath.org/authors/?q=ai:jabin.pierre-emmanuel"Wang, Fei"https://zbmath.org/authors/?q=ai:wang.fei.1|wang.fei.2On the Serrin-type condition on one velocity component for the Navier-Stokes equationshttps://zbmath.org/1472.352682021-11-25T18:46:10.358925Z"Chae, D."https://zbmath.org/authors/?q=ai:chae.dasom|chae.david|chae.dongho|chae.donghyun|chae.dongsuk"Wolf, J."https://zbmath.org/authors/?q=ai:wolf.jorg|wolf.jochen.3|wolf.joseph-a.1|wolf.jana|wolf.jurgen|wolf.jan|wolf.jamison|wolf.judith|wolf.julia|wolf.jesse|wolf.john-p|wolf.joel-l|wolf.jochen.2|wolf.jared-r|wolf.j-benedictA new local regularity criterion for the three-dimensional Navier-Stokes equations is derived. As a consequence a Serrin-type regularity condition imposed on one component of the velocity vector is proved.Boundary stabilizability of the linearized compressible Navier-Stokes system in one dimension by backstepping approachhttps://zbmath.org/1472.352692021-11-25T18:46:10.358925Z"Chowdhury, Shirshendu"https://zbmath.org/authors/?q=ai:chowdhury.shirshendu"Dutta, Rajib"https://zbmath.org/authors/?q=ai:dutta.rajib"Majumdar, Subrata"https://zbmath.org/authors/?q=ai:majumdar.subrataCorrigendum to: ``On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem''https://zbmath.org/1472.352702021-11-25T18:46:10.358925Z"Crispo, F."https://zbmath.org/authors/?q=ai:crispo.francesca"Maremonti, P."https://zbmath.org/authors/?q=ai:maremonti.paoloMisprints in the authors' paper [ibid. 29, No. 4, 1355--1383 (2016, Zbl 1342.35212)] are corrected.Global regularity for solutions of the Navier-Stokes equation sufficiently close to being eigenfunctions of the Laplacianhttps://zbmath.org/1472.352712021-11-25T18:46:10.358925Z"Miller, Evan"https://zbmath.org/authors/?q=ai:miller.evanSummary: In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier-Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian. This estimate improves previous regularity criteria requiring control on the \(\dot{H}^\alpha\) norm of \(u,\) with \(2\leq \alpha <\frac{5}{2}\), to a regularity criterion requiring control on the \(\dot{H}^\alpha\) norm multiplied by the deficit in the interpolation inequality for the embedding of \(\dot{H}^{\alpha -2}\cap\dot{H}^{\alpha}\hookrightarrow \dot{H}^{\alpha -1}\). This regularity criterion suggests, at least heuristically, the possibility of some relationship between potential blowup solutions of the Navier-Stokes equation and the Kolmogorov-Obhukov spectrum in the theory of turbulence.Probabilistic representation for mild solution of the Navier-Stokes equationshttps://zbmath.org/1472.352722021-11-25T18:46:10.358925Z"Olivera, Christian"https://zbmath.org/authors/?q=ai:olivera.christianThe author extends the approach by \textit{P. Constantin} and \textit{G. Iyer} [Commun. Pure Appl. Math. 61, No. 3, 330--345 (2008; Zbl 1156.60048); Ann. Appl. Probab. 21, No. 4, 1466--1492 (2011; Zbl 1246.76018)] to the Navier-Stokes system that involves a probabilistic Lagrangian representation formula making use of stochastic flows. These results are applicable to a more natural class of mild solutions instead of the case of previous works related to classical ones.Global well-posedness and long time decay of fractional Navier-Stokes equations in Fourier-Besov spaceshttps://zbmath.org/1472.352732021-11-25T18:46:10.358925Z"Xiao, Weiliang"https://zbmath.org/authors/?q=ai:xiao.weiliang"Chen, Jiecheng"https://zbmath.org/authors/?q=ai:chen.jiecheng"Fan, Dashan"https://zbmath.org/authors/?q=ai:fan.dashan"Zhou, Xuhuan"https://zbmath.org/authors/?q=ai:zhou.xuhuanSummary: We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spaces \(F \dot{B}_{p, q}^{1 - 2 \beta + 3 / p'}\). Some properties of Fourier-Besov spaces have been discussed, and we prove a general global well-posedness result which covers some recent works in classical Navier-Stokes equations. Particularly, our result is suitable for the critical case \(\beta = 1 / 2\). Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.Decay estimates for three-dimensional Navier-Stokes equations with dampinghttps://zbmath.org/1472.352742021-11-25T18:46:10.358925Z"Zhao, Xiaopeng"https://zbmath.org/authors/?q=ai:zhao.xiaopengThe Navier-Stokes system with the damping term \(|u|^{\beta-1}u\) is studied in the whole space \({\mathbb R}^3\). Decay rates for solutions of the Cauchy problem are derived, together with some auxiliary estimates of negative order Sobolev norms.Low Mach and thin domain limit for the compressible Euler systemhttps://zbmath.org/1472.352752021-11-25T18:46:10.358925Z"Caggio, Matteo"https://zbmath.org/authors/?q=ai:caggio.matteo"Ducomet, Bernard"https://zbmath.org/authors/?q=ai:ducomet.bernard"Nečasová, Šárka"https://zbmath.org/authors/?q=ai:necasova.sarka"Tang, Tong"https://zbmath.org/authors/?q=ai:tang.tongSummary: We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer \(\Omega_{\delta}=(0,\delta)\times\mathbb{R}^2,\delta >0\). In the framework of \textit{dissipative measure-valued solutions}, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and \(\delta\rightarrow 0\).Finite-time singularity formation for an active scalar equationhttps://zbmath.org/1472.352762021-11-25T18:46:10.358925Z"Elgindi, Tarek"https://zbmath.org/authors/?q=ai:elgindi.tarek-mohamed"Ibrahim, Slim"https://zbmath.org/authors/?q=ai:ibrahim.slim"Shen, Shengyi"https://zbmath.org/authors/?q=ai:shen.shengyiStable self-similar blow-up for a family of nonlocal transport equationshttps://zbmath.org/1472.352772021-11-25T18:46:10.358925Z"Elgindi, Tarek M."https://zbmath.org/authors/?q=ai:elgindi.tarek-mohamed"Ghoul, Tej-Eddine"https://zbmath.org/authors/?q=ai:ghoul.tej-eddine"Masmoudi, Nader"https://zbmath.org/authors/?q=ai:masmoudi.naderSummary: We consider a family of nonlocal problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish \textit{stable} self-similar blow-up near a family of known self-similar blow-up solutions.Two-dimensional pseudosteady flows around a sharp cornerhttps://zbmath.org/1472.352782021-11-25T18:46:10.358925Z"Lai, Geng"https://zbmath.org/authors/?q=ai:lai.geng"Sheng, Wancheng"https://zbmath.org/authors/?q=ai:sheng.wanchengSummary: We consider two-dimensional (2D) pseudosteady flows around a sharp corner. This problem can be seen as a 2D Riemann initial and boundary value problem (IBVP) for the compressible Euler system. The initial state is a combination of a uniform flow in one quadrant and vacuum in the remaining domain. The boundary condition on the wall of the sharp corner is a slip boundary condition. By a self-similar transformation, the 2D Riemann IBVP is converted into a boundary value problem (BVP) for the 2D self-similar Euler system. Existence of global piecewise smooth (or Lipshitz-continuous) solutions to the BVP are obtained. One of the main difficulties for the global existence is that the type of the 2D self-similar Euler system is a priori unknown. In order to use the method of characteristic analysis, we establish some a priori estimates for the hyperboliciy of the system. The other main difficulty is that when the uniform flow is sonic or subsonic, the hyperbolic system becomes degenerate at the origin. Moreover, there is a multi-valued singularity at the origin. To solve this degenerate hyperbolic boundary value problem, we establish some uniform interior \(C^{0, 1}\) norm estimates for the solutions of a sequence of regularized hyperbolic boundary value problems, and then use the Arzela-Ascoli theorem and a standard diagonal procedure to construct a global Lipschitz continuous solution. The method used here may also be used to construct continuous solutions of some other degenerate hyperbolic boundary value problems and sonic-supersonic flow problems.Global solutions to the compressible Euler equations with heat transport by convection around Dyson's isothermal affine solutionshttps://zbmath.org/1472.352792021-11-25T18:46:10.358925Z"Rickard, Calum"https://zbmath.org/authors/?q=ai:rickard.calumSummary: Global solutions to the compressible Euler equations with heat transport by convection in the whole space are shown to exist through perturbations of \textit{F. J. Dyson}'s isothermal affine solutions [J. Math. Mech. 18, 91--101 (1968; Zbl 0197.24501)]. This setting presents new difficulties because of the vacuum at infinity behavior of the density. In particular, the perturbation of isothermal motion introduces a Gaussian function into our stability analysis and a novel finite propagation result is proven to handle potentially unbounded terms arising from the presence of the Gaussian. Crucial stabilization-in-time effects of the background motion are mitigated through the use of this finite propagation result however and a careful use of the heat transport formulation in conjunction with new time weight manipulations are used to establish global existence. The heat transport by convection offers unique physical insights into the model and mathematically, we use a controlled spatial perturbation in the analysis of this feature of our system which leads us to exploit source term estimates as part of our techniques.Isentropic approximation of the compressible Euler equations in Besov spaceshttps://zbmath.org/1472.352802021-11-25T18:46:10.358925Z"Wu, Xinglong"https://zbmath.org/authors/?q=ai:wu.xinglongSummary: The article mainly studies the isentropic approximation of the compressible Euler equations in Besov space \(B^s_{2,r}(\mathbb{R}^N)\), provided the initial entropy \( \tilde{S}_0(x)\) changes closing a constant \(\delta\) in the Besov spaces, which extends and improves the result in Sobolev space \(L^2(\mathbb{R}^N)\) by \textit{J. Jia} and \textit{R. Pan} [J. Sci. Comput. 64, No. 3, 745--760 (2015; Zbl 1330.35305)].Integrable systems, multicomponent twisted Heisenberg-Virasoro algebra and its central extensionshttps://zbmath.org/1472.352812021-11-25T18:46:10.358925Z"Wu, Yemo"https://zbmath.org/authors/?q=ai:wu.yemo"Xu, Xiurong"https://zbmath.org/authors/?q=ai:xu.xiurong"Zuo, Dafeng"https://zbmath.org/authors/?q=ai:zuo.dafengSummary: Let \(\mathscr{D}_N\) be the multicomponent twisted Heisenberg-Virasoro algebra. We compute the second continuous cohomology group with coefficients in \(\mathbb{C}\) and study the bihamiltonian Euler equations associated to \(\mathscr{D}_N\) and its central extensions.Time-periodic weak solutions to incompressible generalized Newtonian fluidshttps://zbmath.org/1472.352822021-11-25T18:46:10.358925Z"Abbatiello, Anna"https://zbmath.org/authors/?q=ai:abbatiello.annaSummary: In this study we are interested in the Navier-Stokes-like system for generalized viscous fluids whose viscosity has a power-structure with exponent \(q\). We develop an existence theory of time-periodic three-dimensional flows.Space-time fractional diffusion-advection equation with Caputo derivativehttps://zbmath.org/1472.352832021-11-25T18:46:10.358925Z"Aguilar, José Francisco Gómez"https://zbmath.org/authors/?q=ai:gomez-aguilar.jose-francisco"Hernández, Margarita Miranda"https://zbmath.org/authors/?q=ai:hernandez.margarita-mirandaSummary: An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as \(0 < \beta\), \(\gamma \leq 1\) for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parameters \(\sigma_x\) and \(\sigma_t\) are introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parameters \(\beta\) and \(\gamma\). The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.Kelvin-Voigt equations with anisotropic diffusion, relaxation and damping: blow-up and large time behaviorhttps://zbmath.org/1472.352842021-11-25T18:46:10.358925Z"Antontsev, S."https://zbmath.org/authors/?q=ai:antontsev.stanislav-nikolaevich"De Oliveira, H. B."https://zbmath.org/authors/?q=ai:de-oliveira.hermenegildo-borges"Khompysh, Kh."https://zbmath.org/authors/?q=ai:khompysh.kh|khompysh.khonatbekSummary: A nonlinear initial and boundary-value problem for the Kelvin-Voigt equations with anisotropic diffusion, relaxation and absorption/damping terms is considered in this work. The global and local unique solvability of the problem was established by the first author et al. [J. Math. Anal. Appl. 473, No. 2, 1122--1154 (2019; Zbl 1458.74026)]. In the present work, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients should interact with the problem data for the solutions of this problem display exponential and polynomial time-decays. We also establish the conditions for the solutions of this problem to blow-up in a finite time in three different cases: problem without convection, full anisotropic problem, and the problem with isotropic relaxation.Onsager's conjecture in bounded domains for the conservation of entropy and other companion lawshttps://zbmath.org/1472.352852021-11-25T18:46:10.358925Z"Bardos, C."https://zbmath.org/authors/?q=ai:bardos.claude-w"Gwiazda, P."https://zbmath.org/authors/?q=ai:gwiazda.piotr"Świerczewska-Gwiazda, A."https://zbmath.org/authors/?q=ai:swierczewska-gwiazda.agnieszka"Titi, E. S."https://zbmath.org/authors/?q=ai:titi.edriss-saleh"Wiedemann, E."https://zbmath.org/authors/?q=ai:wiedemann.emilSummary: We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order one-third in the interior of the domain, and if the normal component of the corresponding fluxes tend to zero as one approaches the boundary. This extends various recent results of the authors.Near-critical reflection of internal waveshttps://zbmath.org/1472.352862021-11-25T18:46:10.358925Z"Bianchini, Roberta"https://zbmath.org/authors/?q=ai:bianchini.roberta"Dalibard, Anne-Laure"https://zbmath.org/authors/?q=ai:dalibard.anne-laure"Saint-Raymond, Laure"https://zbmath.org/authors/?q=ai:saint-raymond.laureSummary: Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the slope has the same inclination as the group velocity. We prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the incident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation.Equations of symmetric boundary layer for the Ladyzhenskaya model of a viscous medium in the Crocco variableshttps://zbmath.org/1472.352872021-11-25T18:46:10.358925Z"Bulatova, Regina R."https://zbmath.org/authors/?q=ai:bulatova.regina-r"Samokhin, V. N."https://zbmath.org/authors/?q=ai:samokhin.vyacheslav-n"Chechkin, G. A."https://zbmath.org/authors/?q=ai:chechkin.gregory-aIn this paper authors consider the system of boundary layer equations governing a viscous medium subject to the nonlinear rheological law in the sense of Ladyzhenskaya,
\[
\left\{\begin{array}{l} \nu \left(1+3d\left(\frac{\partial u}{\partial y}\right)^2\right)\frac{\partial^2u}{\partial y^2}-u\frac{\partial u}{\partial x}-v\frac{\partial u}{\partial y}=U\frac{\partial U}{\partial x}\\
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \end{array}\right. \tag{1}
\]
Owing to the use of the Crocco transformation for reducing the system (1) to a single quasilinear equation,
\[
\left\{\begin{array}{ll} \nu(1+3dU^2\omega^2)\omega^2\omega_{\eta\eta}-\eta U\omega_{\xi}+(\eta^2 -1)U_{\xi}\omega_{\eta}-\eta U_{\xi}\omega +6\nu dU^2\omega^{2}_{\eta}\omega^{3}=0 & \hbox{in } 0<\xi<X, 0<\eta<1\\
\omega(\xi,1)=0, \left(\nu(1+3dU^2\omega^2)\omega\omega_{\eta}-v_{0}(\xi)\omega+U_{\xi}\right)|_{\eta=0}=0 & \hbox{in } 0<\xi<X, \end{array}\right. \tag{2}
\]
which becomes possible to study both stationary and nonstationary boundary layers. The authors obtain asymptotic estimates for the solution. In comparison, the von Mises transformation can only handle stationary boundary conditions.Global well-posedness of the 2-d magnetic Prandtl model in the Prandtl-Hartmann regimehttps://zbmath.org/1472.352882021-11-25T18:46:10.358925Z"Chen, Dongxiang"https://zbmath.org/authors/?q=ai:chen.dongxiang"Ren, Siqi"https://zbmath.org/authors/?q=ai:ren.siqi"Wang, Yuxi"https://zbmath.org/authors/?q=ai:wang.yuxi"Zhang, Zhifei"https://zbmath.org/authors/?q=ai:zhang.zhifeiSummary: In this paper, we prove the global well-posedness of the 2-D magnetic Prandtl model in the mixed Prandtl/Hartmann regime when the initial data is a small perturbation of the Hartmann layer in Sobolev space.Rogue wave for the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equationhttps://zbmath.org/1472.352892021-11-25T18:46:10.358925Z"Chen, Hanlin"https://zbmath.org/authors/?q=ai:chen.hanlin"Xu, Zhenhui"https://zbmath.org/authors/?q=ai:xu.zhenhui"Dai, Zhengde"https://zbmath.org/authors/?q=ai:dai.zhengdeSummary: A new method, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solution to nonlinear evolution equation (NEE) is proposed. (3+1)-dimensional Yu-Toda-Sasa-Fukuyama (YTSF) equation is used as an example to illustrate the effectiveness of the suggested method. A new family of two-wave solution, rational breather wave solution, is obtained by extended homoclinic test method, and it is just a rogue wave solution. This result shows rogue wave can come from extreme behavior of breather solitary wave for (3+1)-dimensional nonlinear wave fields.Erratum to: ``On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems''https://zbmath.org/1472.352902021-11-25T18:46:10.358925Z"Della Porta, Francesco"https://zbmath.org/authors/?q=ai:della-porta.francesco"Grasselli, Maurizio"https://zbmath.org/authors/?q=ai:grasselli.maurizioFrom the text: In this note, we want to highlight and correct an error in our paper [ibid. 15, No. 2, 299--317 (2016; Zbl 1334.35226), Prop. 2.4] which has consequences on the proof of [loc. cit., Thm. 6.1]. Referring to [loc. cit.] for the notation, the correct statement in [loc. cit., Prop. 2.4] is that \(\mathbf{u} \in L^2(0, T; [H^1(\Omega)]^d)\) and not \(\mathbf{u} \in L^2(0,T; V_{div})\) as incorrectly written. Therefore we have \(\mathbf{v}(t)=\mathbf{u}(t)- \mathbf{u}_\nu(t)\in [H^1(\Omega)]^d\) for almost any \(t\in (0,T)\) and the boundary trace of \(\mathbf{v}(t)\) is not necessarily zero. Estimates as the one in [loc. cit., Thm. 6.1] are in general difficult to obtain due to the presence of a boundary layer. A common approach to obtain such estimates is to introduce a corrector, so that the difference between the solution and the corrector itself has zero trace. Here we devise a simpler way to obtain an estimate quite similar to the one reported in [loc. cit., Thm. 6.1] without introducing a corrector. However, the order of convergence with respect to \(\nu\) is no longer \(\frac12\).Enhanced diffusivity in perturbed senile reinforced random walk modelshttps://zbmath.org/1472.352912021-11-25T18:46:10.358925Z"Dinh, Thu"https://zbmath.org/authors/?q=ai:dinh.thu"Xin, Jack"https://zbmath.org/authors/?q=ai:xin.jack-xSummary: We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability \(\delta\) of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any \(\delta>0\), with enhanced diffusivity \((\gg O(\delta^2))\) in the small \(\delta\) regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form \(\delta\xi_n\) with \(\xi_n\)'s being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero \(\delta\) limit), with diffusivity between \(O(\frac{1}{|\log\delta|})\) and \(O(\frac{1}{\log |\log\delta|})\). Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as \(O(\log^{-2}\delta)\).Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipationhttps://zbmath.org/1472.352922021-11-25T18:46:10.358925Z"Dong, Boqing"https://zbmath.org/authors/?q=ai:dong.boqing"Wu, Jiahong"https://zbmath.org/authors/?q=ai:wu.jiahong"Xu, Xiaojing"https://zbmath.org/authors/?q=ai:xu.xiaojing"Zhu, Ning"https://zbmath.org/authors/?q=ai:zhu.ningSummary: The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq system helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is \(\mathbb{R}^2\), the stability problem in a Sobolev setting remains open. When the spatial domain is \(\mathbb{T}\times \mathbb{R}\), this paper solves the stability problem and specifies the precise large-time behavior of the perturbation. By decomposing the velocity \(u\) and temperature \(\theta\) into the horizontal average \((\bar{u}, \bar{\theta})\) and the corresponding oscillation \((\widetilde{u},\widetilde{\theta})\), and deriving various anisotropic inequalities, we are able to establish the global stability in the Sobolev space \(H^2\). In addition, we prove that the oscillation \((\widetilde{u},\widetilde{\theta})\) decays exponentially to zero in \(H^1\) and \((u,\theta)\) converges to \((\bar{u}, \bar{\theta})\). This result reflects the stratification phenomenon of buoyancy-driven fluids.Blow up criterion for the 2D full compressible Navier-Stokes equations involving temperature in critical spaceshttps://zbmath.org/1472.352932021-11-25T18:46:10.358925Z"Fan, Jie"https://zbmath.org/authors/?q=ai:fan.jie"Jiu, Quansen"https://zbmath.org/authors/?q=ai:jiu.quansen"Wang, Yanqing"https://zbmath.org/authors/?q=ai:wang.yanqing"Xiao, Yuelong"https://zbmath.org/authors/?q=ai:xiao.yuelongThe authors study the structural stability of conical shocks in the three-dimensional steady isothermal supersonic flows past Lipschitz perturbed cones with small enough vertex angles (less than the critical angle). The problem is symmetrical with respect to the cone's axes, so there remain just two spatial variables. The Bernoulli law serves as the equation of state. The incoming flow left to the obstacle has fixed constant density and supersonic velocity (parallel to the axes of symmetry). The cone is perturbed, with small perturbation in the sense of bounded variation. The main result is existence of global entropy solutions of bounded variation and asymptotic stability of this solution.On the long-time behavior of dissipative solutions to models of non-Newtonian compressible fluidshttps://zbmath.org/1472.352942021-11-25T18:46:10.358925Z"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Kwon, Young-Sam"https://zbmath.org/authors/?q=ai:kwon.young-sam"Novotný, Antonín"https://zbmath.org/authors/?q=ai:novotny.antoninSummary: We identify a class \textit{maximal} dissipative solutions to models of compressible viscous fluids that maximize the energy dissipation rate. Then we show that any maximal dissipative solution approaches an equilibrium state for large times.The vanishing surface tension limit of the Muskat problemhttps://zbmath.org/1472.352952021-11-25T18:46:10.358925Z"Flynn, Patrick T."https://zbmath.org/authors/?q=ai:flynn.patrick-t"Nguyen, Huy Quang"https://zbmath.org/authors/?q=ai:nguyen.huy-quangThe authors investigate the Muskat problem with surface tension. They proved that for any subcritical data satisfying the Rayleigh-Taylor condition, solutions of the Muskat problem with surface tension converge to the unique solution of the Muskat problem without surface tension locally in time. When the initial curvature is square integrable, they obtained the convergence with optimal rate.Dynamics of traveling waves for the perturbed generalized KdV equationhttps://zbmath.org/1472.352962021-11-25T18:46:10.358925Z"Ge, Jianjiang"https://zbmath.org/authors/?q=ai:ge.jianjiang"Wu, Ranchao"https://zbmath.org/authors/?q=ai:wu.ranchao"Du, Zengji"https://zbmath.org/authors/?q=ai:du.zengjiSummary: This paper is devoted to the existence of traveling, solitary and periodic waves for the perturbed generalized KdV by applying geometric singular perturbation, differential manifold theory and the regular perturbation analysis of Hamiltonian systems. Under the assumptions that the distributed delay kernel is the strong general one and the average delay is sufficiently small, traveling, solitary and periodic waves are shown to exist in the perturbed system. It is further proved that the wave speed is decreasing by analyzing the ratio of Abelian integrals, and we analyze these functions by using the theory of analytic functions and algebraic geometry. Moreover, the upper and lower bounds of the limit wave speed are presented. The relationship between wavelength and wave speed of traveling waves is also established.Erratum to: ``Volume viscosity and internal energy relaxation: symmetrization and Chapman-Enskog expansion''https://zbmath.org/1472.352972021-11-25T18:46:10.358925Z"Giovangigli, Vincent"https://zbmath.org/authors/?q=ai:giovangigli.vincent"Yong, Wen-An"https://zbmath.org/authors/?q=ai:yong.wen-anErratum to the authors' paper [ibid. 8, No. 1, 79--116 (2015; Zbl 1310.35196)].Partial regularity of suitable weak solutions of the Navier-Stokes-Planck-Nernst-Poisson equationhttps://zbmath.org/1472.352982021-11-25T18:46:10.358925Z"Gong, Huajun"https://zbmath.org/authors/?q=ai:gong.huajun"Wang, Changyou"https://zbmath.org/authors/?q=ai:wang.changyou"Zhang, Xiaotao"https://zbmath.org/authors/?q=ai:zhang.xiaotaoLarge-time behavior of magnetohydrodynamics with temperature-dependent heat-conductivityhttps://zbmath.org/1472.352992021-11-25T18:46:10.358925Z"Huang, Bin"https://zbmath.org/authors/?q=ai:huang.bin"Shi, Xiaoding"https://zbmath.org/authors/?q=ai:shi.xiaoding"Sun, Ying"https://zbmath.org/authors/?q=ai:sun.yingSummary: For the strong solutions to the equations of a planar magnetohydrodynamic compressible flow with the heat conductivity proportional to a nonnegative power of the temperature, we first prove that both the specific volume and the temperature are proved to be bounded from below and above independently of time. Then, we also show that the global strong solution is nonlinearly exponentially stable as time tends to infinity. This is the first result obtaining the exponential stability behavior of strong solutions to the equations of a planar magnetohydrodynamic compressible flow without any smallness conditions on the data. Our result can be regarded as a natural generalization of the previous ones for the compressible Navier-Stokes system to MHD system with either constant heat-conductivity or nonlinear and temperature-depending heat-conductivity. As a direct consequence, it is shown that the global strong solution to the constant heat-conductivity MHD system whose existence is obtained by \textit{A. V. Kazhikhov} [``A priori estimates for the solutions of equations of magnetic gas dynamics'' (Russian), in: Boundary-value problems for equations of mathematical physics. Krasnoyarsk. 84--94 (1987)] is nonlinearly exponentially stable.Weak solutions of Hopf type to 2D Maxwell flows with infinite number of relaxation timeshttps://zbmath.org/1472.353002021-11-25T18:46:10.358925Z"Karazeeva, N. A."https://zbmath.org/authors/?q=ai:karazeeva.n-aSummary: A system of equations describing the motion of fluids of Maxwell type is considered:
\[
\frac{\partial }{\partial t}v +v \cdot \nabla v -\underset{0}{\overset{t}{\int }}K\left(t-\tau \right) d\tau +\nabla p=f\left(x,t\right), \quad\mathrm{div}v =0.
\]
Here \(K(t)\) is an exponential series \( K(t)=\sum \limits_{s=1}^{\infty }{\beta}_s{e}^{-{\alpha}_st}\). The existence of a weak solution for the initial boundary value problem
\[
v (x,0)=v_0(x),\quad v \cdot n|_{\partial \Omega }=0,\quad rot v |_{\partial \Omega }=0
\]
is proved.Some regularity criteria for the 3D generalized Navier-Stokes equationshttps://zbmath.org/1472.353012021-11-25T18:46:10.358925Z"Kim, Jae-Myoung"https://zbmath.org/authors/?q=ai:kim.jaemyoungGeneralizations and improvements of Serrin's type regularity criterion for weak solutions of the three-dimensional Navier-Stokes system with the dissipation defined by fractional Laplacians are shown. These are conditions expressed in terms of the vorticity of solutions using Lorentz spaces.Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical conditionhttps://zbmath.org/1472.353022021-11-25T18:46:10.358925Z"Kobayashi, Takayuki"https://zbmath.org/authors/?q=ai:kobayashi.takayuki"Tsuda, Kazuyuki"https://zbmath.org/authors/?q=ai:tsuda.kazuyukiSummary: In this research, we study the global existence of solutions to the compressible Navier-Stokes-Korteweg system around a constant state. This system describes liquid-vapor type two-phase flow with a phase transition with diffuse interface. Previous works assume that pressure is a monotone function for change of density similarly to the usual compressible Navier-Stokes system. On the other hand, due to phase transition the pressure is in fact non-monotone function, and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. We show that global \(L^2\) solutions are available for the critical case of small data, whose momentum is in its derivative form, and obtain parabolic type decay rate of the solutions. This is proved based on the decomposition of solutions to a low frequency part and a high frequency part.Weak solutions for a sixth order Cahn-Hilliard type equation with degenerate mobilityhttps://zbmath.org/1472.353032021-11-25T18:46:10.358925Z"Liu, Aibo"https://zbmath.org/authors/?q=ai:liu.aibo"Liu, Changchun"https://zbmath.org/authors/?q=ai:liu.chein-shanSummary: We study an initial-boundary problem for a sixth order Cahn-Hilliard type equation, which arises in oil-water-surfactant mixtures. An existence result for the problem with a concentration dependent diffusional mobility in three space dimensions is presented.Well-posedness of the MHD boundary layer system in Gevrey function space without structural assumptionhttps://zbmath.org/1472.353042021-11-25T18:46:10.358925Z"Li, Wei-Xi"https://zbmath.org/authors/?q=ai:li.weixi"Yang, Tong"https://zbmath.org/authors/?q=ai:yang.tong.1|yang.tongStability of blow-up solution for the two component Camassa-Holm equationshttps://zbmath.org/1472.353052021-11-25T18:46:10.358925Z"Li, Xintao"https://zbmath.org/authors/?q=ai:li.xintao"Huang, Shoujun"https://zbmath.org/authors/?q=ai:huang.shoujun"Yan, Weiping"https://zbmath.org/authors/?q=ai:yan.weipingSummary: This paper studies the wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the two component Camassa-Holm equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler's equation in the shallow water regime.Exact solutions and conservation laws of the Drinfel'd-Sokolov-Wilson systemhttps://zbmath.org/1472.353062021-11-25T18:46:10.358925Z"Matjila, Catherine"https://zbmath.org/authors/?q=ai:matjila.catherine"Muatjetjeja, Ben"https://zbmath.org/authors/?q=ai:muatjetjeja.ben"Khalique, Chaudry Masood"https://zbmath.org/authors/?q=ai:khalique.chaudry-masoodSummary: We study the Drinfel'd-Sokolov-Wilson system, which was introduced as a model of water waves. Firstly we obtain exact solutions of this system using the \((G' / G)\)-expansion method. In addition to exact solutions we also construct conservation laws for the underlying system using Noether's approach.Global existence and the decay of solutions to the Prandtl system with small analytic datahttps://zbmath.org/1472.353072021-11-25T18:46:10.358925Z"Paicu, Marius"https://zbmath.org/authors/?q=ai:paicu.marius"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.3Summary: In this paper, we prove the global existence and the large time decay estimate of solutions to Prandtl system with small initial data, which is analytical in the tangential variable. The key ingredient used in the proof is to derive a sufficiently fast decay-in-time estimate of some weighted analytic energy estimate to a quantity, which consists of a linear combination of the tangential velocity with its primitive one, and which basically controls the evolution of the analytical radius to the solutions. Our result can be viewed as a global-in-time Cauchy-Kowalevsakya result for the Prandtl system with small analytical data, which in particular improves the previous result in [\textit{M. Ignatova} and \textit{V. Vicol}, ibid. 220, No. 2, 809--848 (2016; Zbl 1334.35238)] concerning the almost global well-posedness of a two-dimensional Prandtl system.Computing multiple solutions of topology optimization problemshttps://zbmath.org/1472.353082021-11-25T18:46:10.358925Z"Papadopoulos, Ioannis P. A."https://zbmath.org/authors/?q=ai:papadopoulos.ioannis-p-a"Farrell, Patrick E."https://zbmath.org/authors/?q=ai:farrell.patrick-e"Surowiec, Thomas M."https://zbmath.org/authors/?q=ai:surowiec.thomasOn evolutionary inverse problems for mathematical models of heat and mass transferhttps://zbmath.org/1472.353092021-11-25T18:46:10.358925Z"Pyatkov, Sergeĭ Grigor'evich"https://zbmath.org/authors/?q=ai:pyatkov.sergei-gSummary: This article is a survey. The results on well-posedness of inverse problems for mathematical models of heat and mass transfer are presented. The unknowns are the coefficients of a system or the right-hand side (the source function). The overdetermination conditions are values of a solution of some manifolds or integrals of a solution with weight over the spatial domain. Two classes of mathematical models are considered. The former includes the Navier-Stokes system, the parabolic equations for the temperature of a fluid, and the parabolic system for concentrations of admixtures. The right-hand side of the system for concentrations is unknown and characterizes the volumetric density of sources of admixtures in a fluid. The unknown functions depend on time and some part of spacial variables and occur in the right-hand side of the parabolic system for concentrations. The latter class is just a parabolic system of equations, where the unknowns occur in the right-hand side and the system as coefficients. The well-posedness questions for these problems are examined, in particular, existence and uniqueness theorems as well as stability estimates for solutions are exposed.Hydrodynamic entrance region in a flat porous channel with a pressure head isothermal laminar flow of a Newtonian mediumhttps://zbmath.org/1472.353102021-11-25T18:46:10.358925Z"Ryazhskikh, Aleksandr Viktorovich"https://zbmath.org/authors/?q=ai:ryazhskikh.aleksandr-viktorovich"Nikolenko, Aleksandr Vladimirovich"https://zbmath.org/authors/?q=ai:nikolenko.aleksandr-vladimirovich"Konovalov, Dmitriĭ Al'bertovich"https://zbmath.org/authors/?q=ai:konovalov.dmitrii-albertovich"Ryazhskih, Viktor Ivanovich"https://zbmath.org/authors/?q=ai:ryazhskih.viktor-ivanovich"Keller, Alevtina Viktorovna"https://zbmath.org/authors/?q=ai:keller.alevtina-viktorovnaSummary: The problem of the hydrodynamic initial section of an isothermal pressure laminar flow of a Newtonian fluid in a horizontal flat porous channel of semi-infinite length, formulated in the initial-boundary formulation for the Darcy-Brinkman equation with partial consideration of the convective component, provided that the pressure depends only on the axial coordinate, is analytically solved. For a channel without a porous matrix, the results correlate with the classical data. An explicit relation was proposed for calculating the length of the hydrodynamic initial section, which does not contradict the results based on macroscopic boundary layer concepts.Eulerian dynamics in multidimensions with radial symmetryhttps://zbmath.org/1472.353112021-11-25T18:46:10.358925Z"Tan, Changhui"https://zbmath.org/authors/?q=ai:tan.changhuiA remark on the regularity criterion for the 3D Boussinesq equations involving the pressure gradienthttps://zbmath.org/1472.353122021-11-25T18:46:10.358925Z"Zhang, Zujin"https://zbmath.org/authors/?q=ai:zhang.zujinSummary: We consider the three-dimensional Boussinesq equations and obtain a regularity criterion involving the pressure gradient in the Morrey-Companato space \(M_{p, q}\). This extends and improves the result of \textit{S. Gala} [Appl. Anal. 92, No. 1, 96--103 (2013; Zbl 1284.35313)] for the Navier-Stokes equations.Remarks on the regularity criteria for the axisymmetric MHD systemhttps://zbmath.org/1472.353132021-11-25T18:46:10.358925Z"Zhang, Zujin"https://zbmath.org/authors/?q=ai:zhang.zujin"Zhang, Yali"https://zbmath.org/authors/?q=ai:zhang.yaliSummary: We show several weighted regularity criteria for the axisymmetric solutions to the three-dimensional magnetohydrodynamic equations, involving \(u^r, u^z; u^r, b^r\), \(\partial_ru^z, \partial_rb^z\); \(u^r,b^r, \partial_zu^\theta\), \(\partial_zb^\theta; u^r, b^r\), \(\partial_ru^\theta,\partial_rb^\theta\); or \(\omega^\theta\), where \(u^r,u^\theta,u^z\) are the angular, swirl and axial components of the velocity respectively and \(\omega^\theta\) denotes the swirl component of the vorticity.Global strong solution to the nonhomogeneous Bénard system with large initial data and vacuumhttps://zbmath.org/1472.353142021-11-25T18:46:10.358925Z"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinSummary: We establish a unique global strong solution for nonhomogeneous Bénard system with zero density at infinity on the whole two-dimensional (2D) space. In particular, the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Our method relies heavily on the structure of the system under consideration and spatial dimension.Global strong solution of nonhomogeneous Bénard system with large initial data and vacuum in a bounded domainhttps://zbmath.org/1472.353152021-11-25T18:46:10.358925Z"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinSummary: We study an initial boundary value problem of two-dimensional nonhomogeneous Bénard system with nonnegative density. We derive the global existence of a unique strong solution. In particular, the initial data can be arbitrarily large.On the generalization of Moyal equation for an arbitrary linear quantizationhttps://zbmath.org/1472.353162021-11-25T18:46:10.358925Z"Borisov, Leonid A."https://zbmath.org/authors/?q=ai:borisov.leonid-a"Orlov, Yuriy N."https://zbmath.org/authors/?q=ai:orlov.yurii-nLocal and global solutions to the \(O(3)\)-sigma model with the Maxwell and the Chern-Simons gauges in \(\mathbb{R}^{1 + 1} \)https://zbmath.org/1472.353172021-11-25T18:46:10.358925Z"Jin, Guanghui"https://zbmath.org/authors/?q=ai:jin.guanghui"Moon, Bora"https://zbmath.org/authors/?q=ai:moon.boraIn this paper the authors study the well-posedness of the \((1 +1)\)-dimensional Maxwell-Chern- Simons gauged \(O(3)\)-sigma model (MCSS) with the Lorenz gauge condition. The Lagrangian density of the \((2+1)\)-dimensional MCSS has the form
\[
\mathcal L=-(1/4\rho )F_{\mu\nu }F^{\mu\nu }+ (\kappa /4)\epsilon^{\mu\nu\lambda } F_{\mu\nu }A_{\lambda}+(1/2)D_{\mu }\phi\cdot D^{\mu }\phi + 1/(2\rho ) \partial_{\mu }N\partial^{\mu }N-V(\phi ,N),
\]
where it is used the Minkowski metric , \(\phi=(\phi_1,\phi_2,\phi_3): \mathbb{R}^{2+1}\to S^2\) is a vector field, \(N:\mathbb{R}^{2+1}\to\mathbb{R}\) is a neutral scalar field and \(A_{\lambda }:\mathbb{R}^{2+1}\to\mathbb{R}\) (\(\lambda =0,1,2\)) is the gauge field, \(F_{\mu\nu }\) is the curvature tensor, and \(D_{\mu }\) is the covariant derivative. One may accept \(\rho =1\). By a standard way it is obtained the Euler-Lagrange system of partial differential equations. There are two possible natural asymptotic conditions for \( (\varphi , N, A_{\mu }, B) \) to make the total energy finite: (i) nontopological boundary condition, and (ii) topological boundary condition. Here the authors consider the nontopological boundary condition. The Euler-Lagrange system is invariant under the gauge transformations. It is accepted the gauge which is fixed by adopting the Lorenz gauge condition \(\partial_0 A_0 - \partial_1 A_1 = 0\). Under the Lorenz gauge condition, the Cauchý problem for the \((1+1)\)-dimensional MCSS system along with initial data can be rewritten in a proper form. In the first part the authors study the global well-posedness of a solution in a high regular space. The main statement is that if the initial data
\[
\varphi_0, \ A_{\mu } (0,\cdot ) = a_{0\mu }, \ B(0,\cdot )=b_0, \ N(0,\cdot )=n_0\in H^2,
\]
\[
\partial_t\phi (0,\cdot )= \varphi_1, \ \partial_tA_{\mu }(0,\cdot )=a_{1\mu }, \ \partial_tB(0,\cdot )= b_1, \ \partial_tN(0,\cdot )=n_1\in H^1,
\]
and certain constraint hold, then the Cauchý problem for MCSS system has a unique, global in time solution \((\phi , N,A_{\mu },B)\) whose components belong to \(C([0,\infty ),H^2(\mathbb{R}))\cap C^1 ([0,\infty ), H^1(\mathbb{R}))\). Also it is shown that existence of a global solution with corresponding finite total energy.
Moreover, the authors discuss the local well-posedness below the energy regularity by exploiting null form and wave-Sobolev space.Born approximation and sequence for hyperbolic equationshttps://zbmath.org/1472.353182021-11-25T18:46:10.358925Z"Lin, Ching-Lung"https://zbmath.org/authors/?q=ai:lin.ching-lung"Lin, Liren"https://zbmath.org/authors/?q=ai:lin.liren"Nakamura, Gen"https://zbmath.org/authors/?q=ai:nakamura.genSummary: The Born approximation and the Born sequence are considered for hyperbolic equations when we perturb their leading parts. The Born approximation is a finite successive approximation such as the finite terms Neumann series for the solution of a hyperbolic equation in terms of the smallness of the perturbation and if the successive approximation is infinitely many times, then we have the Born series. Due to the so called regularity loss for solutions of hyperbolic equations, we need to assume that data such as the inhomogeneous term of the equation, Cauchy datum and boundary datum are \(\mathit{C}^\infty \), and also they satisfy the compatibility condition of any order in order to define the Born series. Otherwise we need to smooth each term of the Born series. The convergence of the Born series and the Born series with smoothing are very natural questions to be asked. Also giving an estimate of approximating the solution for finite terms Born series is also an important question in practice. The aims of this paper are to discuss about these questions. We would like to emphasize that we found a small improvement in the usual energy estimate for solutions of an initial value problem for a hyperbolic equation, which is very useful for our aims. Since the estimate of approximation is only giving the worst estimate for the approximation, we also provide some numerical studies on these questions which are very suggestive for further theoretical studies on the Born approximation for hyperbolic equations.Scaling limits of bosonic ground states, from many-body to non-linear Schrödingerhttps://zbmath.org/1472.353192021-11-25T18:46:10.358925Z"Rougerie, Nicolas"https://zbmath.org/authors/?q=ai:rougerie.nicolasThis article gives a comprehensive review of the ground state of bosonic systems. These systems are studied in terms of mean-field approximations. This work contains five comprehensive chapters, an index and over three-hundred references.Jarzynski equality for conditional stochastic workhttps://zbmath.org/1472.353202021-11-25T18:46:10.358925Z"Sone, Akira"https://zbmath.org/authors/?q=ai:sone.akira"Deffner, Sebastian"https://zbmath.org/authors/?q=ai:deffner.sebastianSummary: It has been established that the \textit{inclusive work} for classical, Hamiltonian dynamics is equivalent to the two-time energy measurement paradigm in isolated quantum systems. However, a plethora of other notions of quantum work has emerged, and thus the natural question arises whether any other quantum notion can provide motivation for purely classical considerations. In the present analysis, we propose the \textit{conditional stochastic work} for classical, Hamiltonian dynamics, which is inspired by the one-time measurement approach. This novel notion is built upon the change of expectation value of the energy conditioned on the initial energy surface. As main results, we obtain a generalized Jarzynski equality and a sharper maximum work theorem, which account for how non-adiabatic the process is. Our findings are illustrated with the parametric harmonic oscillator.Evolution by Schrödinger equation of Aharonov-Berry superoscillations in centrifugal potentialhttps://zbmath.org/1472.353212021-11-25T18:46:10.358925Z"Colombo, F."https://zbmath.org/authors/?q=ai:colombo.fabrizio"Gantner, J."https://zbmath.org/authors/?q=ai:gantner.jonathan"Struppa, D. C."https://zbmath.org/authors/?q=ai:struppa.daniele-carloSummary: In recent years, we have investigated the evolution of superoscillations under Schrödinger equation with non-singular potentials. In all those cases, we have shown that superoscillations persist in time. In this paper, we investigate the centrifugal potential, which is a singular potential, and we show that the techniques developed to study the evolution of superoscillations in the case of the Schrödinger equation with a quadratic Hamiltonian apply to this setting. We also specify, in the case of the centrifugal potential, the notion of super-shift of the limit solution, a fact explained in the last section of this paper. It then becomes apparent that superoscillations are just a particular case of super-shift.On the one dimensional Dirac equation with potentialhttps://zbmath.org/1472.353222021-11-25T18:46:10.358925Z"Erdoğan, M. Burak"https://zbmath.org/authors/?q=ai:erdogan.mehmet-burak"Green, William R."https://zbmath.org/authors/?q=ai:green.william-rSummary: We investigate \(L^1\to L^\infty\) dispersive estimates for the one dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural \(t^{-\frac{1}{2}}\) decay rate, which may be improved to \(t^{-\frac{3}{2}}\) at the cost of spatial weights when the thresholds are regular. We classify the structure of threshold obstructions, showing that there is at most a one dimensional space at each threshold. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate, and satisfies the faster weighted bound except for a piece of rank at most two, one per threshold. Further, we prove high energy dispersive bounds that are near optimal with respect to the required smoothness of the initial data. To do so we use a variant of a high energy argument that was originally developed to study Kato smoothing estimates for magnetic Schrödinger operators. This method has never been used before to obtain \(L^1\to L^\infty\) estimates. As a consequence of our analysis we prove a uniform limiting absorption principle, Strichartz estimates, and prove the existence of an eigenvalue free region for the one dimensional Dirac operator with a non-self-adjoint potential.Semiclassical evolution with low regularityhttps://zbmath.org/1472.353232021-11-25T18:46:10.358925Z"Golse, François"https://zbmath.org/authors/?q=ai:golse.francois"Paul, Thierry"https://zbmath.org/authors/?q=ai:paul.thierrySummary: We prove semiclassical estimates for the Schrödinger-von Neumann evolution with \(C^{1,1}\) potentials and density matrices whose square root have either Wigner functions with low regularity independent of the dimension, or matrix elements between Hermite functions having long range decay. The estimates are settled in different weak topologies and apply to initial density operators whose square root have Wigner functions 7 times differentiable, independently of the dimension. They also apply to the \(N\)-body quantum dynamics uniformly in \(N\) and to concentrating pure and mixed states without any regularity assumption. In an appendix, we finally estimate the dependence in the dimension of the constant appearing on the Calderón-Vaillancourt Theorem.Uncertainty principle, minimal escape velocities, and observability inequalities for Schrödinger equationshttps://zbmath.org/1472.353242021-11-25T18:46:10.358925Z"Huang, Shanlin"https://zbmath.org/authors/?q=ai:huang.shanlin"Soffer, Avy"https://zbmath.org/authors/?q=ai:soffer.avrahamSummary: We develop a new abstract derivation of the observability inequalities at two points in time for Schrödinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator \(H\) on \(L^2(\mathbb{R}^n)\). In the second step we use results on asymptotic behavior of \(e^{-itH} \), in particular, minimal velocity estimates introduced by \textit{I. M. Sigal} and the second author [Invent. Math. 99, No. 1, 115--143 (1990; Zbl 0702.35197)]. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schrödinger equation.Exact analytical solutions: physical and/or mathematical validityhttps://zbmath.org/1472.353252021-11-25T18:46:10.358925Z"Kamdoum-Tamo, P. H."https://zbmath.org/authors/?q=ai:kamdoum-tamo.p-h"Tala-Tebue, E."https://zbmath.org/authors/?q=ai:tala-tebue.eric"Kenfack-Jiotsa, A."https://zbmath.org/authors/?q=ai:kenfack-jiotsa.aurelien"Kofane, T. C."https://zbmath.org/authors/?q=ai:kofane.timoleon-crepinSummary: In this work, we use the alternative \((G'/G)\)-expansion method, the sech method, the tanh method and the Painleve truncated approach to find solutions of the modified complex Ginzburg-Landau equation. We show that any mathematically acceptable solution is not necessarily physically suitable. Among the two types of obtained solutions, there is a category with null infinite branches, for which no direct numerical simulation can be carried out. This type of solutions is however mathematically well-grounded. The second type concerns new solutions with infinite non-zero branches. For this second type, direct numerical simulations are performed to show that they are physically valid.Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: high dimensionshttps://zbmath.org/1472.353262021-11-25T18:46:10.358925Z"Li, Ze"https://zbmath.org/authors/?q=ai:li.zeSummary: In this paper, we prove that the Schrödinger map flows from \(\mathbb{R}^d\) with \(d \geq 3\) to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. This is a companion work of the authors' previous paper [``Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: energy critical case'', Preprint. \url{arXiv:1811.10924}] where the energy critical case \(d = 2\) was solved. In the first part of this paper, for heat flows from \(\mathbb{R}^d\) \((d \geq 3)\) to Riemannian manifolds with small data in critical Sobolev spaces, we prove the decay estimates of moving frame dependent quantities in the caloric gauge setting, which is of independent interest and may be applied to other problems. In the second part, with a key bootstrap-iteration scheme in our previous work [loc. cit.], we apply these decay estimates to the study of Schrödinger map flows by choosing caloric gauge. This work with our previous work solves the open problem raised by \textit{D. Tataru} [Am. J. Math. 123, No. 1, 37--77 (2001; Zbl 0979.35100)].Multi-scale analysis for wave problems with disparate propagation speedshttps://zbmath.org/1472.353272021-11-25T18:46:10.358925Z"Bostan, Mihaï"https://zbmath.org/authors/?q=ai:bostan.mihaiSummary: The subject matter of this work concerns the asymptotic behavior of the wave problems, when the propagation speed in one direction is much larger than in the other directions. We establish weak and strong convergence results. We appeal to homogenization arguments, based on average operators with respect to unitary groups.Transport equations and perturbations of boundary conditionshttps://zbmath.org/1472.353282021-11-25T18:46:10.358925Z"Tyran-Kamińska, Marta"https://zbmath.org/authors/?q=ai:tyran-kaminska.martaSummary: We provide a new perturbation theorem for substochastic semigroups on abstract AL spaces extending Kato's perturbation theorem to nondensely defined operators. We show how it can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions. We give particular examples to illustrate our results.Remarks on Riemann and Ricci solitons in \((\alpha,\beta)\)-contact metric manifoldshttps://zbmath.org/1472.353292021-11-25T18:46:10.358925Z"Blaga, Adara M."https://zbmath.org/authors/?q=ai:blaga.adara-monica"Lațcu, Dan Radu"https://zbmath.org/authors/?q=ai:latcu.dan-raduSummary: We study almost Riemann solitons and almost Ricci solitons in an \((\alpha,\beta)\)-contact metric manifold satisfying some Ricci symmetry conditions, treating the case when the potential vector field of the soliton is pointwise collinear with the structure vector field.Whitham equations and phase shifts for the Korteweg-de Vries equationhttps://zbmath.org/1472.353302021-11-25T18:46:10.358925Z"Ablowitz, Mark J."https://zbmath.org/authors/?q=ai:ablowitz.mark-j"Cole, Justin T."https://zbmath.org/authors/?q=ai:cole.justin-t"Rumanov, Igor"https://zbmath.org/authors/?q=ai:rumanov.igorSummary: The semi-classical Korteweg-de Vries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis, Whitham theory is constructed to the higher order. This allows the order one phase and the complete leading-order solution to be obtained; the results are confirmed by extensive numerical calculations.Binary Darboux transformation for a negative-order AKNS equationhttps://zbmath.org/1472.353312021-11-25T18:46:10.358925Z"Amjad, Z."https://zbmath.org/authors/?q=ai:amjad.z"Khan, D."https://zbmath.org/authors/?q=ai:khan.d-n|khan.dil-faraz|khan.diba|khan.debashis|khan.dost-muhammad|khan.dawarSummary: We study a binary Darboux transformation for a negative-order AKNS equation. Iterating the transformation, we obtain \(N\)-fold quasi-Grammian solutions expressed in terms of quasideterminants. In some simple cases, we construct explicit solutions of the studied equation with nonvanishing and vanishing backgrounds including bright and dark breathers, a soliton, and solutions with one and two humps.A proof of validity for multiphase Whitham modulation theoryhttps://zbmath.org/1472.353322021-11-25T18:46:10.358925Z"Bridges, Thomas J."https://zbmath.org/authors/?q=ai:bridges.thomas-j"Kostianko, Anna"https://zbmath.org/authors/?q=ai:kostianko.anna"Schneider, Guido"https://zbmath.org/authors/?q=ai:schneider.guido.1|schneider.guidoSummary: It is proved that approximations which are obtained as solutions of the multiphase Whitham modulation equations stay close to solutions of the original equation on a natural time scale. The class of nonlinear wave equations chosen for the starting point is coupled nonlinear Schrödinger equations. These equations are not in general integrable, but they have an explicit family of multiphase wavetrains that generate multiphase Whitham equations, which may be elliptic, hyperbolic, or of mixed type. Due to the change of type, the function space set-up is based on Gevrey spaces with initial data analytic in a strip in the complex plane. In these spaces a Cauchy-Kowalevskaya-like existence and uniqueness theorem is proved. Building on this theorem and higher-order approximations to Whitham theory, a rigorous comparison of solutions, of the coupled nonlinear Schrödinger equations and the multiphase Whitham modulation equations, is obtained.Stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noisehttps://zbmath.org/1472.353332021-11-25T18:46:10.358925Z"Cheng, Shuilin"https://zbmath.org/authors/?q=ai:cheng.shuilin"Guo, Yantao"https://zbmath.org/authors/?q=ai:guo.yantao"Tang, Yanbin"https://zbmath.org/authors/?q=ai:tang.yanbinSummary: This paper considers a stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise. We first prove the existence and uniqueness of strong probabilistic solution of an initial-boundary value problem with homogeneous Dirichlet boundary conditions. Then we give an asymptotic behavior of the solution.Restrictions on the existence of a canonical system flow hierarchyhttps://zbmath.org/1472.353342021-11-25T18:46:10.358925Z"Hur, Injo"https://zbmath.org/authors/?q=ai:hur.injo"Ong, Darren C."https://zbmath.org/authors/?q=ai:ong.darren-cSummary: The KdV hierarchy is a family of evolutions on a Schrödinger operator that preserves its spectrum. Canonical systems are a generalization of Schrödinger operators, that nevertheless share many features with Schrödinger operators. Since this is a very natural generalization, one would expect that it would also be straightforward to build a hierarchy of isospectral evolutions on canonical systems analogous to the KdV hierarchy. Surprisingly, we show that there are many obstructions to constructing a hierarchy of flows on canonical systems that obeys the standard assumptions of the KdV hierarchy. This suggests that we need a more sophisticated approach to develop such a hierarchy, if it is indeed possible to do so.On an integrable multi-dimensionally consistent \(2n + 2n\)-dimensional heavenly-type equationhttps://zbmath.org/1472.353352021-11-25T18:46:10.358925Z"Konopelchenko, B. G."https://zbmath.org/authors/?q=ai:konopelchenko.boris-g"Schief, W. K."https://zbmath.org/authors/?q=ai:schief.wolfgang-karlSummary: Based on the commutativity of scalar vector fields, an algebraic scheme is developed which leads to a privileged multi-dimensionally consistent \(2n + 2n\)-dimensional integrable partial differential equation with the associated eigenfunction constituting an infinitesimal symmetry. The `universal' character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalizing to higher dimensions a great variety of well-known integrable equations such as the dispersionless Kadomtsev-Petviashvili and Hirota equations and various avatars of the heavenly equation governing self-dual Einstein spaces.An integrable lattice hierarchy associated with a \(4 \times 4\) matrix spectral problem: \(N\)-fold Darboux transformation and dynamical propertieshttps://zbmath.org/1472.353362021-11-25T18:46:10.358925Z"Liu, Ling"https://zbmath.org/authors/?q=ai:liu.ling"Wen, Xiao-Yong"https://zbmath.org/authors/?q=ai:wen.xiaoyong"Liu, Nan"https://zbmath.org/authors/?q=ai:liu.nan"Jiang, Tao"https://zbmath.org/authors/?q=ai:jiang.tao.1|jiang.tao|jiang.tao.2"Yuan, Jin-Yun"https://zbmath.org/authors/?q=ai:yuan.jin-yun|yuan.jinyunSummary: A new integrable hierarchy related to a \(4 \times 4\) matrix isospectral problem is proposed, in which the semi-discrete version of the coupled KdV system is the first member. Subsequently the \(N\)-fold Darboux transformation for the second member in the obtained hierarchy is constructed. As applications, the explicitly exact solutions to the equation in three cases of different seed solutions are discussed and their graphs are showed to analyze the corresponding dynamical properties. Meanwhile some numerical simulations are given to illustrate these properties.Application of the homotopy analysis method for solving the variable coefficient KdV-Burgers equationhttps://zbmath.org/1472.353372021-11-25T18:46:10.358925Z"Lu, Dianchen"https://zbmath.org/authors/?q=ai:lu.dianchen"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie.2|liu.jie.4|liu.jie.7|liu.jie.1|liu.jie.5|liu.jie.3|liu.jieSummary: The homotopy analysis method is applied to solve the variable coefficient KdV-Burgers equation. With the aid of generalized elliptic method and Fourier's transform method, the approximate solutions of double periodic form are obtained. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. The results indicate that this method is efficient for the nonlinear models with the dissipative terms and variable coefficients.Solving nonlinear non-local problems using positive square-root operatorshttps://zbmath.org/1472.353382021-11-25T18:46:10.358925Z"Montagu, E. L."https://zbmath.org/authors/?q=ai:montagu.e-l"Norbury, John"https://zbmath.org/authors/?q=ai:norbury.john-wSummary: A non-constructive existence theory for certain operator equations \[L u = D u,\] using the substitution \(u = B^{\frac{1}{2}} \xi\) with \(B = L^{-1} \), is developed, where \(L\) is a linear operator (in a suitable Banach space) and \(D\) is a homogeneous nonlinear operator such that \(D \lambda u = \lambda^{} \alpha D u\) for all \(\lambda \geq 0\) and some \(\alpha \in \mathbb{R}, \alpha \neq \) ~1. This theory is based on the positive-operator approach of Krasnosel'skii. The method has the advantage of being able to tackle the nonlinear right-hand side \(D\) in cases where conventional operator techniques fail. By placing the requirement that the operator \(B\) must have a positive square root, it is possible to avoid the usual regularity condition on either the mapping \(D\) or its Fréchet derivative. The technique can be applied in the case of elliptic PDE problems, and we show the existence of solitary waves for a generalization of Benjamin's fluid dynamics problem.Correction to: ``Uniqueness and stability of global conservative solutions for the modified coupled Camassa-Holm system''https://zbmath.org/1472.353392021-11-25T18:46:10.358925Z"Pan, Shihang"https://zbmath.org/authors/?q=ai:pan.shihang"Zhou, Shouming"https://zbmath.org/authors/?q=ai:zhou.shouming"Zhang, Baoshuai"https://zbmath.org/authors/?q=ai:zhang.baoshuaiCorrection to the authors' paper [ibid. 100, No. 6, 1301--1326 (2021; Zbl 1468.35177)].The Cauchy problem for a fifth-order dispersive equationhttps://zbmath.org/1472.353402021-11-25T18:46:10.358925Z"Wang, Hongjun"https://zbmath.org/authors/?q=ai:wang.hongjun"Liu, Yongqi"https://zbmath.org/authors/?q=ai:liu.yongqi"Chen, Yongqiang"https://zbmath.org/authors/?q=ai:chen.yongqiangSummary: This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev space \(H^s(\mathbb{R})\) with \(s \geq 1 / 4\). We also establish the ill-posedness for the initial data in \(H^s(\mathbb{R})\) with \(s < 1 / 4\). Thus, the regularity requirement for the fifth-order dispersive equations \(s \geq 1 / 4\) is sharp.Time-dependent defects in integrable soliton equationshttps://zbmath.org/1472.353412021-11-25T18:46:10.358925Z"Xia, Baoqiang"https://zbmath.org/authors/?q=ai:xia.baoqiang"Zhou, Ruguang"https://zbmath.org/authors/?q=ai:zhou.ruguangSummary: We study (1 + 1)-dimensional integrable soliton equations with time-dependent defects located at \(x = c(t)\), where \(c(t)\) is a function of class \(C^1\). We define the defect condition as a Bäcklund transformation evaluated at \(x = c(t)\) in space rather than over the full line. We show that such a defect condition does not spoil the integrability of the system. We also study soliton solutions that can meet the defect for the system. An interesting discovery is that the defect system admits peaked soliton solutions.Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor setshttps://zbmath.org/1472.353422021-11-25T18:46:10.358925Z"Yang, Ai-Min"https://zbmath.org/authors/?q=ai:yang.aimin"Zhang, Yu-Zhu"https://zbmath.org/authors/?q=ai:zhang.yuzhu"Cattani, Carlo"https://zbmath.org/authors/?q=ai:cattani.carlo"Xie, Gong-Nan"https://zbmath.org/authors/?q=ai:xie.gongnan"Rashidi, Mohammad Mehdi"https://zbmath.org/authors/?q=ai:rashidi.mohammad-mehdi"Zhou, Yi-Jun"https://zbmath.org/authors/?q=ai:zhou.yijun"Yang, Xiao-Jun"https://zbmath.org/authors/?q=ai:yang.xiao-junSummary: We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.On differential equations derived from the pseudospherical surfaceshttps://zbmath.org/1472.353432021-11-25T18:46:10.358925Z"Yang, Hongwei"https://zbmath.org/authors/?q=ai:yang.hongwei"Wang, Xiangrong"https://zbmath.org/authors/?q=ai:wang.xiangrong"Yin, Baoshu"https://zbmath.org/authors/?q=ai:yin.baoshuSummary: We construct two metric tensor fields; by means of these metric tensor fields, sinh-Gordon equation and elliptic sinh-Gordon equation are obtained, which describe pseudospherical surfaces of constant negative Riemann curvature scalar \(\sigma\) = \(-\)2, \(\sigma\) = \(-\)1, respectively. By employing the Bäcklund transformation, nonlinear superposition formulas of sinh-Gordon equation and elliptic sinh-Gordon equation are derived; various new exact solutions of the equations are obtained.The generalized projective Riccati equations method for solving nonlinear evolution equations in mathematical physicshttps://zbmath.org/1472.353442021-11-25T18:46:10.358925Z"Zayed, E. M. E."https://zbmath.org/authors/?q=ai:zayed.elsayed-m-e"Alurrfi, K. A. E."https://zbmath.org/authors/?q=ai:alurrfi.khaled-a-eSummary: We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.Dynamics of a 3D Benjamin-Bona-Mahony equations with sublinear operatorhttps://zbmath.org/1472.353452021-11-25T18:46:10.358925Z"Zhao, Mingxia"https://zbmath.org/authors/?q=ai:zhao.mingxia"Yang, Xin-Guang"https://zbmath.org/authors/?q=ai:yang.xinguang"Yan, Xingjie"https://zbmath.org/authors/?q=ai:yan.xingjie"Cui, Xiaona"https://zbmath.org/authors/?q=ai:cui.xiaonaSummary: This paper is concerned with the tempered pullback dynamics for a three dimensional Benjamin-Bona-Mahony equations with sublinear operator on bounded domain, which describes the long time behavior for long waves model in shallow water with friction. By virtue of a new retarded Gronwall inequality, and using the energy equation method from \textit{J. M. Ball} [Discrete Contin. Dyn. Syst. 10, No. 1--2, 31--52 (2004; Zbl 1056.37084)] to achieve asymptotic compactness for solution process, the minimal family of pullback attractors has been obtained, which reduces a single trajectory under a sufficient condition.Local well-posedness for a type of periodic fifth-order Korteweg-de Vries equations without nonlinear dispersive termhttps://zbmath.org/1472.353462021-11-25T18:46:10.358925Z"Zhou, Deqin"https://zbmath.org/authors/?q=ai:zhou.deqinSummary: We consider the Cauchy problem of the fifth-order Korteweg-de Vries (KdV) equations without nonlinear dispersive term
\[
\partial_t u - \partial_x^5 u + b_0 u \partial_x u + b_1 \partial_x (\partial_x u)^2 = 0, \; (t, x) \in \mathbb{R} \times \mathbb{T}. \tag{\(0.1\)}
\]
Recently, \textit{T. Kappeler} and \textit{J. Molnar} [Sel. Math., New Ser. 24, No. 2, 1479--1526 (2018; Zbl 1393.37078); Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35, No. 1, 101--160 (2018; Zbl 1406.37050)] proved that the fifth-order KdV equation with nonlinear dispersive term and Hamiltonian structure is globally well-posed in \(H^s(\mathbb{T})\) with \(s \geq 0\). Without the nonlinear dispersive term, Equation (0.1) is not integrable, and Kappeler-Molnar's approach is not valid. Using the idea of modifying Bourgain space, we prove that Equation (0.1) is locally well-posed in \(H^s(\mathbb{T})\) with \(s \geq \frac{5}{8}\).Inverse scattering transform for the focusing nonlinear Schrödinger equation with counterpropagating flowshttps://zbmath.org/1472.353472021-11-25T18:46:10.358925Z"Biondini, Gino"https://zbmath.org/authors/?q=ai:biondini.gino"Lottes, Jonathan"https://zbmath.org/authors/?q=ai:lottes.jonathan"Mantzavinos, Dionyssios"https://zbmath.org/authors/?q=ai:mantzavinos.dionyssiosSummary: The inverse scattering transform for the focusing nonlinear Schrödinger equation is presented for a general class of initial conditions whose asymptotic behavior at infinity consists of counterpropagating waves. The formulation takes into account the branched nature of the two asymptotic eigenvalues of the associated scattering problem. The Jost eigenfunctions and scattering coefficients are defined explicitly as single-valued functions on the complex plane with jump discontinuities along certain branch cuts. The analyticity properties, symmetries, discrete spectrum, asymptotics, and behavior at the branch points are discussed explicitly. The inverse problem is formulated as a matrix Riemann-Hilbert problem with poles. Reductions to all cases previously discussed in the literature are explicitly discussed. The scattering data associated to a few special cases consisting of physically relevant Riemann problems are explicitly computed.On the inhomogeneous NLS with inverse-square potentialhttps://zbmath.org/1472.353482021-11-25T18:46:10.358925Z"Campos, Luccas"https://zbmath.org/authors/?q=ai:campos.luccas"Guzmán, Carlos M."https://zbmath.org/authors/?q=ai:guzman.carlos-mSummary: We consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in \(\mathbb{R}^N\)
\[
iu_t-\mathcal{L}_au+\lambda |x|^{-b}|u|^\alpha u=0,\quad \mathcal{L}_a=-\Delta +\frac{a}{|x|^2},
\]
where \(\lambda=\pm 1\), \(\alpha,b>0\) and \(a>-\frac{(N-2)^2}{4}\). We first establish sufficient conditions for global existence and blow-up in \(H^1_a(\mathbb{R}^N)\) for \(\lambda=1\), using a Gagliardo-Nirenberg-type estimate. In the sequel, we study local and global well-posedness in \(H^1_a(\mathbb{R}^N)\) in the \(H^1\)-subcritical case, applying the standard Strichartz estimates combined with the fixed point argument. The key to do that is to establish good estimates on the nonlinearity. Making use of these estimates, we also show a scattering criterion and construct a wave operator in \(H^1_a(\mathbb{R}^N)\), for the mass-supercritical and energy-subcritical case.The existence and uniqueness result for a relativistic nonlinear Schrödinger equationhttps://zbmath.org/1472.353492021-11-25T18:46:10.358925Z"Cheng, Yongkuan"https://zbmath.org/authors/?q=ai:cheng.yongkuan"Yang, Jun"https://zbmath.org/authors/?q=ai:yang.jun.1|yang.jun.3|yang.jun.2|yang.junSummary: We study the existence and uniqueness of positive solutions for a class of quasilinear elliptic equations. This model has been proposed in the self-channeling of a high-power ultrashort laser in matter.Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponentshttps://zbmath.org/1472.353502021-11-25T18:46:10.358925Z"Chen, Jianhua"https://zbmath.org/authors/?q=ai:chen.jianhua"Huang, Xianjiu"https://zbmath.org/authors/?q=ai:huang.xianjiu"Qin, Dongdong"https://zbmath.org/authors/?q=ai:qin.dongdong"Cheng, Bitao"https://zbmath.org/authors/?q=ai:cheng.bitaoSummary: In this paper, we study the following generalized quasilinear Schrödinger equation
\[
-\operatorname{div}(\varepsilon^2g^2(u)\nabla u)+\varepsilon^2g(u) g^\prime(u)|\nabla u|^2+V(x)u=K(x)|u|^{p-2}u+|u|^{22^\ast-2}u,\quad x\in\mathbb{R}^{N},
\]
where \(N\geqslant 3\), \(\varepsilon>0\), \(4<p<22^\ast\), \(g\in\mathcal{C}^1(\mathbb{R},\mathbb{R}^+)\), \(V\in\mathcal{C}(\mathbb{R}^{N})\cap L^\infty(\mathbb{R}^{N})\) has a positive global minimum, and \(K\in\mathcal{C}(\mathbb{R}^{N})\cap L^\infty(\mathbb{R}^N)\) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik-Schnirelmann theory, we also prove the existence of multiple solutions.Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulationhttps://zbmath.org/1472.353512021-11-25T18:46:10.358925Z"Cingolani, S."https://zbmath.org/authors/?q=ai:cingolani.silvia"Gallo, M."https://zbmath.org/authors/?q=ai:gallo.mariano|gallo.mose|gallo.mirko|gallo.michele"Tanaka, K."https://zbmath.org/authors/?q=ai:tanaka.kazunagaDynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equationshttps://zbmath.org/1472.353522021-11-25T18:46:10.358925Z"Dinh, Van Duong"https://zbmath.org/authors/?q=ai:dinh.van-duongThe author studies the Cauchy problem for a class of the fourth-order nonlinear Schrödinger equations \[ \left\{ \begin{array}{rcl} i\partial_t u - \Delta^2 u + \mu \Delta u &=& \pm |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N, \\
u(0,x)&=& u_0(x), \end{array} \right.\tag{4NLS}\] where \(u: \mathbb{R} \times \mathbb{R}^N \rightarrow \mathbb{C}\), \(u_0: \mathbb{R}^N \rightarrow \mathbb{C}\), \(\mu \in \mathbb{R}\), and \(\alpha>0\).
The equation (4NLS) has formally the conservation of mass and energy
\begin{align*}
M(u(t))&=\int |u(t,x)|^2 dx = M(u_0), \tag{Mass} \\
E_\mu(u(t)) &= \frac{1}{2} \int |\Delta u(t,x)|^2 dx +\frac{\mu}{2} \int |\nabla u(t,x)|^2 dx \pm \frac{1}{\alpha+2} \int |u(t,x)|^{\alpha+2} dx = E_\mu(u_0). \tag{Energy}
\end{align*}
Let
\[
\gamma_c:= \frac{N}{2} -\frac{4}{\alpha}, \qquad \sigma_c:= \frac{2-\gamma_c}{\gamma_c} = \frac{8-(N-4)\alpha}{N\alpha-8}, \qquad \alpha^*:= \left\{ \begin{array}{cl} \frac{8}{N-4} &\text{if } N\geq 5, \\
\infty &\text{if } N\leq 4. \end{array} \right.
\]
The first result is the following energy scattering below the ground state for the focusing problem (4NLS). Theorem theo-scat. Let \(N\geq 2\), \(\mu\geq 0\), and \(\frac{8}{N}<\alpha<\alpha^*\). Let \(u_0 \in H^2\) be radially symmetric and satisfy \begin{align*} E_\mu(u_0) [M(u_0)]^{\sigma_c} &< E_0(Q) [M(Q)]^{\sigma_c}, \tag{cond-ener} \\
\|\Delta u_0\|_{L^2} \|u_0\|^{\sigma_c}_{L^2} & < \|\Delta Q\|_{L^2} \|Q\|^{\sigma_c}_{L^2}. \tag{cond-grad-gwp} \end{align*} Then the corresponding solution to the focusing problem (4NLS) exists globally in time and scatters in \(H^2\) in both directions, i.e. there exist \(u_\pm \in H^2\) such that
\[
\lim_{t\rightarrow \pm \infty} \|u(t) - e^{-it(\Delta^2-\mu \Delta)} u_\pm\|_{H^2} =0.
\]
The second result concerns the finite time blow-up in the mass and energy intercritical case.
Theorem theo-blup-inter. Let \(N\geq 2\), \(\mu\geq 0\), \(\frac{8}{N}<\alpha<\alpha^*\), and \(\alpha \leq 8\). Let \(u_0 \in H^2\) be radially symmetric satisfying (cond-ener) and \begin{align*} \|\Delta u_0\|_{L^2} \|u_0\|^{\sigma_c}_{L^2} > \|\Delta Q\|_{L^2} \|Q\|^{\sigma_c}_{L^2}.\tag{cond-grad-blup} \end{align*} Then the corresponding solution to the focusing problem (4NLS) blows up in finite time.
The third result is the following finite time blow-up in the energy critical case.
Theorem theo-blup-ener. Let \(N\geq 5\), \(\mu\geq 0\), and \(\alpha=\frac{8}{N-4}\). Let \(u_0 \in H^2\) be radially symmetric satisfying \begin{align*} E_\mu(u_0) &< E_0(W), \tag{cond-ener-ener} \\
\|\Delta u_0\|_{L^2} &> \|\Delta W\|_{L^2}, \tag{cond-grad-ener} \end{align*} where \(W\) is the unique non-negative radial solution to \begin{align*} \Delta^2 W - |W|^{\frac{8}{N-4}} W=0. \tag{ell-equ-W}\end{align*} Then the corresponding solution to the focusing problem (4NLS) blows up in finite time.On nonlinear Schrödinger equations with repulsive inverse-power potentialshttps://zbmath.org/1472.353532021-11-25T18:46:10.358925Z"Dinh, Van Duong"https://zbmath.org/authors/?q=ai:dinh.van-duongSummary: In this paper, we consider the Cauchy problem for nonlinear Schrödinger equations with repulsive inverse-power potentials
\[
i\partial_t u+\Delta u-c|x|^{-\sigma}u=\pm |u|^{\alpha}u,\quad c> 0.
\]
We study the local and global well-posedness, finite time blow-up and scattering in the energy space for the equation. These results extend a recent work of \textit{C. Miao} et al. [``Nonlinear Schrödinger equation with Coulomb potential'', Preprint, \url{arXiv:1809.06685}] to a general class of inverse-power potentials and higher dimensions.Counterexamples to \(L^p\) collapsing estimateshttps://zbmath.org/1472.353542021-11-25T18:46:10.358925Z"Du, Xiumin"https://zbmath.org/authors/?q=ai:du.xiumin"Machedon, Matei"https://zbmath.org/authors/?q=ai:machedon.mateiSummary: We show that certain \(L^p\) space-time estimates for generalized density matrices which have been used by several authors in recent years to study equations of BBGKY or Hartree-Fock type, do not have non-trivial \(L^p L^p\) generalizations.Blowup dynamics for mass critical half-wave equation in 3Dhttps://zbmath.org/1472.353552021-11-25T18:46:10.358925Z"Georgiev, Vladimir"https://zbmath.org/authors/?q=ai:georgiev.vladimir-s"Li, Yuan"https://zbmath.org/authors/?q=ai:li.yuan|li.yuan.2|li.yuan.3|li.yuan.1Summary: We consider the half-wave equation \(i u_t = D u - | u |^{\frac{ 2}{ 3}} u\) in three dimensions and in the mass critical. For initial data \(u( t_0, x) = u_0(x) \in H_{\mathrm{rad}}^{1 / 2 + \delta}( \mathbb{R}^3)\) with radial symmetry, we construct a new class of the radial blowup solutions with the blow up rate \(\| D^{\frac{ 1}{ 2}} u ( t ) \|_2 \sim \frac{ C ( u_0 )}{ | t |}\) as \(t \to 0^-\).Potential well theory for the derivative nonlinear Schrödinger equationhttps://zbmath.org/1472.353562021-11-25T18:46:10.358925Z"Hayashi, Masayuki"https://zbmath.org/authors/?q=ai:hayashi.masayukiSummary: We consider the following nonlinear Schrödinger equation of derivative type:
\[
i\,\partial_t u+\partial_x^2u+i|u|^2\,\partial_xu+b|u|^4u=0,\quad (t,x)\in\mathbb{R}\times\mathbb{R},\, b\in\mathbb{R}.
\]
If \(b=0\), this equation is known as a standard derivative nonlinear Schrödinger equation (DNLS), which is mass-critical and completely integrable. The equation above can be considered as a generalized equation of DNLS while preserving mass-criticality and Hamiltonian structure. For DNLS it is known that if the initial data \(u_0\in H^1(\mathbb{R})\) satisfies the mass condition \(\|u_0\|_{L^2}^2 <4\pi\), the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation above for general \(b\in\mathbb{R}\), which corresponds exactly to the \(4\pi\)-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass-threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both the \(4\pi\)-mass condition and algebraic solitons.Patterns of water in lighthttps://zbmath.org/1472.353572021-11-25T18:46:10.358925Z"Horikis, Theodoros P."https://zbmath.org/authors/?q=ai:horikis.theodoros-p"Frantzeskakis, Dimitrios J."https://zbmath.org/authors/?q=ai:frantzeskakis.dimitri-jSummary: The intricate patterns emerging from the interactions between soliton stripes of a two-dimensional defocusing nonlinear Schrödinger (NLS) model with a non-local nonlinearity are considered. We show that, for sufficiently strong non-locality, the model is asymptotically reduced to a Kadomtsev-Petviashvilli-II (KPII) equation, which is a common model arising in the description of shallow water waves, as such patterns of water may indeed exist in light (this non-local NLS finds applications in nonlinear optics, modelling beam propagation in media featuring thermal nonlinearities, in plasmas, and in nematic liquid crystals). This way, approximate antidark soliton solutions of the NLS model are constructed from the stable KPII line solitons. By means of direct numerical simulations, we demonstrate that non-resonant and resonant two- and three-antidark NLS stripe soliton interactions give rise to wave configurations that are found in the context of the KPII equation. Thus, our study indicates that patterns which are usually observed in water can also be found in optics.Dynamics of nearly parallel vortex filaments for the Gross-Pitaevskii equationhttps://zbmath.org/1472.353582021-11-25T18:46:10.358925Z"Jerrard, R. L."https://zbmath.org/authors/?q=ai:jerrard.robert-leon"Smets, D."https://zbmath.org/authors/?q=ai:smets.didierSummary: \textit{R. Klein} et al. [J. Fluid Mech. 288, 201--248 (1995; Zbl 0846.76015)] have formally derived a simplified asymptotic motion law for the evolution of nearly parallel vortex filaments in the context of the three dimensional Euler equation for incompressible fluids. In the present work, we rigorously derive the corresponding asymptotic motion law in the context of the Gross-Pitaevskii equation.Corrigendum to: ``On Schrödinger systems with cubic dissipative nonlinearities of derivative type''https://zbmath.org/1472.353592021-11-25T18:46:10.358925Z"Li, Chunhua"https://zbmath.org/authors/?q=ai:li.chunhua"Sunagawa, Hideaki"https://zbmath.org/authors/?q=ai:sunagawa.hideakiFrom the text: In this note, we correct errors which have occurred in the proof of lemma 5.2 in our paper [ibid. 29, No. 5, 1537--1563 (2016, Zbl 1338.35411)]. There is no change in the statement of this lemma, so this correction does not affect the main results or any other parts in [loc. cit].Asymptotic behavior of the coupled nonlinear Schrödinger lattice systemhttps://zbmath.org/1472.353602021-11-25T18:46:10.358925Z"Li, Hengyan"https://zbmath.org/authors/?q=ai:li.hengyan"Zhao, Xin"https://zbmath.org/authors/?q=ai:zhao.xinSummary: This paper studies asymptotic behavior of solutions for the coupled nonlinear Schrödinger lattice system. We obtain the existence and stability of compact attractor by means of tail estimates method and finite-dimensional approximations.Asymptotics for a class of fractional coupled Schrödinger systemshttps://zbmath.org/1472.353612021-11-25T18:46:10.358925Z"Saanouni, T."https://zbmath.org/authors/?q=ai:saanouni.tarekThe paper investigates the Cauchy problem for a fractional Schrödinger system with a coupled power-type non-linearity. A sharp threshold of global existence versus blow-up dichotomy was obtained. Moreover, using the potential-well method, the existence of non global solutions is proved. Finally, instability by blow-up of mass-super-critical and energy sub-critical standing waves is obtained.Energy scattering for radial focusing inhomogeneous bi-harmonic Schrödinger equationshttps://zbmath.org/1472.353622021-11-25T18:46:10.358925Z"Saanouni, Tarek"https://zbmath.org/authors/?q=ai:saanouni.tarekSummary: This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation
\[
i\dot{u}+\Delta^2 u+F(x,u)=0.
\]
Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to [\textit{B. Dodson} and \textit{J. Murphy}, Proc. Am. Math. Soc. 145, No. 11, 4859--4867 (2017; Zbl 1373.35287)].Growth of Sobolev norms for coupled lowest Landau level equationshttps://zbmath.org/1472.353632021-11-25T18:46:10.358925Z"Schwinte, Valentin"https://zbmath.org/authors/?q=ai:schwinte.valentin"Thomann, Laurent"https://zbmath.org/authors/?q=ai:thomann.laurentSummary: We study coupled systems of nonlinear lowest Landau level equations, for which we prove global existence results with polynomial bounds on the possible growth of Sobolev norms of the solutions. We also exhibit explicit unbounded trajectories, which show that these bounds are optimal.Remarks on blow-up criteria for the derivative nonlinear Schrödinger equation under the optimal threshold settinghttps://zbmath.org/1472.353642021-11-25T18:46:10.358925Z"Takaoka, Hideo"https://zbmath.org/authors/?q=ai:takaoka.hideoSummary: We study the Cauchy problem of the mass critical nonlinear Schrödinger equation with derivative with the \(4 \pi\) mass. One has the global well-posedness in \(H^1\) whenever ``the mass is strictly less than \(4 \pi \)'' or whenever ``the mass is equal to \(4 \pi\) and the momentum is strictly less than zero''. In this paper, by the concentration compact principle as originally done by \textit{C. E. Kenig} and \textit{F. Merle} [Invent. Math. 166, No. 3, 645--675 (2006; Zbl 1115.35125)], we obtain the limiting profile of blow up solutions with the critical \(4 \pi\) mass.Energy thresholds of blow-up for the Hartree equation with a focusing subcritical perturbationhttps://zbmath.org/1472.353652021-11-25T18:46:10.358925Z"Tian, Shuai"https://zbmath.org/authors/?q=ai:tian.shuai"Yang, Ying"https://zbmath.org/authors/?q=ai:yang.ying"Zhou, Rui"https://zbmath.org/authors/?q=ai:zhou.rui"Zhu, Shihui"https://zbmath.org/authors/?q=ai:zhu.shihuiSummary: This paper studies the blow-up solutions for the Schrödinger equation with a Hartree-type nonlinearity together with a power-type subcritical perturbation. The precisely sharp energy thresholds for blow-up and global existence are obtained by analyzing potential well structures for associated functionals.Erratum to: ``Triggered fronts in the complex Ginzburg Landau equation''https://zbmath.org/1472.353662021-11-25T18:46:10.358925Z"Goh, Ryan"https://zbmath.org/authors/?q=ai:goh.ryan-n"Scheel, Arnd"https://zbmath.org/authors/?q=ai:scheel.arndErratum to the authors' paper [ibid. 24, No. 1, 117--144 (2014; Zbl 1297.35224)].Random attractors for stochastic Ginzburg-Landau equation on unbounded domainshttps://zbmath.org/1472.353672021-11-25T18:46:10.358925Z"Lu, Qiuying"https://zbmath.org/authors/?q=ai:lu.qiuying"Deng, Guifeng"https://zbmath.org/authors/?q=ai:deng.guifeng"Zhang, Weipeng"https://zbmath.org/authors/?q=ai:zhang.weipengSummary: We prove the existence of a pullback attractor in \(\mathbb{L}^2\)\((\mathbb{R}^n)\) for the stochastic Ginzburg-Landau equation with additive noise on the entire \(n\)-dimensional space \(\mathbb{R}^n\). We show that the stochastic Ginzburg-Landau equation with additive noise can be recast as a random dynamical system. We demonstrate that the system possesses a unique \(\mathcal{D}\)-random attractor, for which the asymptotic compactness is established by the method of uniform estimates on the tails of its solutions.Existence and limiting behavior of min-max solutions of the Ginzburg-Landau equations on compact manifoldshttps://zbmath.org/1472.353682021-11-25T18:46:10.358925Z"Stern, Daniel"https://zbmath.org/authors/?q=ai:stern.danielSummary: We use a natural two-parameter min-max construction to produce critical points of the Ginzburg-Landau functionals on a compact Riemannian manifold of dimension \(\geq 2\). We investigate the limiting behavior of these critical points as \(\varepsilon \to 0\), and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable \((n-2)\)-varifold as \(\epsilon \to 0\), suggesting connections to the min-max construction of minimal \((n-2)\)-submanifolds.Non-convex \(\ell_p\) regularization for sparse reconstruction of electrical impedance tomographyhttps://zbmath.org/1472.353692021-11-25T18:46:10.358925Z"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing|wang.jing.13|wang.jing.1|wang.jing.11|wang.jing.14|wang.jing.2|wang.jing.3|wang.jing.6|wang.jing.17|wang.jing.16|wang.jing.5|wang.jing.15Summary: This work is to investigate the image reconstruction of electrical impedance tomography from the electrical measurements made on an object's surface. An \(\ell_p\)-norm \((0<p<1)\) sparsity-promoting regularization is considered to deal with the fully non-linear electrical impedance tomography problem, and a novel type of smoothing gradient-type iteration scheme is introduced. To avoid the difficulty in calculating its gradient in the optimization process, a smoothing Huber potential function is utilized to approximate the \(\ell_p\)-norm penalty. We then propose the smoothing algorithm in the general frame and establish that any accumulation point of the generated iteration sequence is a first-order stationary point of the original problem. Furthermore, one iteration scheme based on the homotopy perturbation technology is derived to find the minimizers of the Huberized approximated objective function. Numerical experiments show that non-convex \(\ell_p\)-norm sparsity-promoting regularization improves the spatial resolution and is more robust with respect to noise, in comparison with \(\ell_p\)-norm regularization.Retraction notice to: ``Small perturbation of a surface: full Maxwell's equations''https://zbmath.org/1472.353702021-11-25T18:46:10.358925Z"Kassraoui, Jihed"https://zbmath.org/authors/?q=ai:kassraoui.jihedFrom Editor's note: Subsequent to publication, we were made aware that this paper was a duplicate publication of the following paper: [\textit{A. Khelifi} and \textit{S. Boujemaa}, J. Math. Anal. Appl. 444, No. 2, 1721--1738 (2016; Zbl 1351.35210)].Numerical scheme for electromagnetic scattering on perturbed periodic inhomogeneous media and reconstruction of the perturbationhttps://zbmath.org/1472.353712021-11-25T18:46:10.358925Z"Konschin, Alexander"https://zbmath.org/authors/?q=ai:konschin.alexanderWeak solutions of the relativistic Vlasov-Maxwell system with external currentshttps://zbmath.org/1472.353722021-11-25T18:46:10.358925Z"Weber, Jörg"https://zbmath.org/authors/?q=ai:weber.jorgThe paper studies the Vlasov-Maxwell system and their weak solutions when the plasma is contained in a bounded domain \(\Omega\) while the electromagnetic fields are induced by external currents outside the container. Here, the main contribution is that boundary conditions on \(E\) and \(H\) fields is not set to perfect electric conductor (PEC) at the boundaries of container. This is much more difficult problem but makes it possible for interaction of the fields inside container with currents in the control coils.
The author uses a method similar to [\textit{Y. Guo}, Commun. Math. Phys. 154, No. 2, 245--263 (1993; Zbl 0787.35072)] to identity the proper sets of test functions and function spaces for particle densities as well as \(E\) and \(H\) fields. The first of main results of the article is about the existence of the weak solutions for such a Vlasov-Maxwell system. The second major outcome is the examination of the divergence equations and demonstration of the redundancy of the \(E\) field based on the proposed weak formulation.
The proposed weak formulation is used in a separate article by the same author [SIAM J. Math. Anal. 52, No. 3, 2895--2929 (2020; Zbl 1448.35499)] to find optimal external currents that control the plasma particles.Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimensionhttps://zbmath.org/1472.353732021-11-25T18:46:10.358925Z"Tu, Son N. T."https://zbmath.org/authors/?q=ai:tu.son-n-tSummary: Let \(u^\varepsilon\) and \(u\) be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence \(\mathcal{O}(\varepsilon)\) of \(u^{\varepsilon}\to u\) as \(\varepsilon\to 0^+\) for a large class of convex Hamiltonians \(H(x,y,p)\) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension \(n=1\).On solutions for a class of Kirchhoff systems involving critical growth in \(\mathbb{R}^2\)https://zbmath.org/1472.353742021-11-25T18:46:10.358925Z"de Albuquerque, J. C."https://zbmath.org/authors/?q=ai:de-albuquerque.jose-carlos"Do Ó., J. M."https://zbmath.org/authors/?q=ai:do-o.joao-m-bezerra"dos Santos, E. O."https://zbmath.org/authors/?q=ai:dos-santos.e-o"Severo, U. B."https://zbmath.org/authors/?q=ai:severo.uberlandio-batistaSummary: In this work we study the existence of solutions for the following class of elliptic systems involving Kirchhoff equations in the plane:
\[
\begin{cases}
m(\|u\|^2)[-\Delta u+u]=\lambda f(u,v),\quad & x\in\mathbb{R}^2, \\
\ell(\|v\|^2)[-\Delta v+v]=\lambda g(u,v), \quad & x\in\mathbb{R}^2,
\end{cases}
\]
where \(\lambda>0\) is a parameter, \(m,\ell:[0,+\infty)\to[0,+\infty)\) are Kirchhoff-type functions, \(\|\cdot\|\) denotes the usual norm of the Sobolev space \(H^1(\mathbb{R}^2)\) and the nonlinear terms \(f\) and \(g\) have exponential critical growth of Trudinger-Moser type. Moreover, when \(f\) and \(g\) are odd functions, we prove that the number of solutions increases when the parameter \(\lambda\) becomes large.Retraction notice to: ``Stability and global attractors for thermoelastic Bresse system''https://zbmath.org/1472.353752021-11-25T18:46:10.358925Z"Ma, Zhiyong"https://zbmath.org/authors/?q=ai:ma.zhiyongFrom the text: The Editor and Publishers Taylor \& Francis are retracting the above article from publication in Applicable Analysis. This paper was previously published in online
only form as part of Taylor \& Francis' iFirst service. Date of online publication 15th April
2011, \url{http://dx.doi.org/10.1080/00036811.2011.559463}. This article is now available at
\url{https://doi.org/10.1080/00036811.2011.559463}.
The author failed to cite, reference, or properly acknowledge their previous work published in Advances in Difference Equations detailed below nor properly identify or quote extracts from that work which subsequently appeared in, and appeared original to the article (now retracted) in, Applicable Analysis:
[\textit{Z. Ma}, Adv. Difference Equ. 2010, Article ID 748789, 15~p. (2010; Zbl 1204.35162)].Solutions to a phase-field model for martensitic phase transformations driven by configurational forceshttps://zbmath.org/1472.353762021-11-25T18:46:10.358925Z"Wu, Fan"https://zbmath.org/authors/?q=ai:wu.fan"Bian, Xingzhi"https://zbmath.org/authors/?q=ai:bian.xingzhi"Zhao, Lixian"https://zbmath.org/authors/?q=ai:zhao.lixianSummary: We study the existence of weak solutions to an initial-boundary value problem for a new phase-field model, which consists of a degenerate parabolic equation coupled to linear elasticity equations. This model is used to describe the evolution of interfaces in elastically deformable solid materials which moves by configurational forces, such as martensitic phase transformations in shape memory alloys. The existence proof is valid only for one-dimensional case.How to smooth a crinkled map of space-time: Uhlenbeck compactness for \(L^\infty\) connections and optimal regularity for general relativistic shock waves by the Reintjes-Temple equationshttps://zbmath.org/1472.353772021-11-25T18:46:10.358925Z"Reintjes, Moritz"https://zbmath.org/authors/?q=ai:reintjes.moritz"Temple, Blake"https://zbmath.org/authors/?q=ai:temple.blakeSummary: We present the authors' new theory of the RT-equations (`regularity transformation' or `Reintjes-Temple' equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections \(\Gamma\) to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem \(( \Gamma )\). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in \(L^p\) spaces. The theory and results announced in this paper apply to arbitrary \(L^\infty\) connections on the tangent bundle \(T \mathcal{M}\) of arbitrary manifolds \(\mathcal{M} \), including Lorentzian manifolds of general relativity.Topological sensitivity analysis for a three-dimensional parabolic type problemhttps://zbmath.org/1472.353782021-11-25T18:46:10.358925Z"Ghezaiel, Emna"https://zbmath.org/authors/?q=ai:ghezaiel.emna"Abdelwahed, Mohamed"https://zbmath.org/authors/?q=ai:abdelwahed.mohamed"Chorfi, Nejmeddine"https://zbmath.org/authors/?q=ai:chorfi.nejmeddine"Hassine, Maatoug"https://zbmath.org/authors/?q=ai:hassine.maatougSummary: This work focuses on the topological sensitivity analysis of a three-dimensional parabolic type problem. The considered application model is described by the heat equation. We derive a new topological asymptotic expansion valid for various shape functions and geometric perturbations of arbitrary form. The used approach is based on a rigorous mathematical framework describing and analyzing the asymptotic behavior of the perturbed temperature field.Exact solutions for the conformable space-time fractional Zeldovich equation with time-dependent coefficientshttps://zbmath.org/1472.353792021-11-25T18:46:10.358925Z"Injrou, Sami"https://zbmath.org/authors/?q=ai:injrou.samiSummary: The aim of this paper is to improve a sub-equation method to solve the space-time fractional Zeldovich equation with time-dependent coefficients involving the conformable fractional derivative. As result, we obtain three families of solutions including the hyperbolic, trigonometric, and rational solutions. These solutions may be helpful to explain several phenomena in chemistry, including the combustion process. The study shows that the used method is effective and reliable and can be utilized as a substitution to construct new solutions of different types of nonlinear conformable fractional partial differential equations (NFPDEs) with variable coefficients.Global zero-relaxation limit of the non-isentropic Euler-Poisson system for ion dynamicshttps://zbmath.org/1472.353802021-11-25T18:46:10.358925Z"Feng, Yuehong"https://zbmath.org/authors/?q=ai:feng.yuehong"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin.5|li.xin.12|li.xin.10|li.xin.15|li.xin.14|li.xin.1|li.xin.11|li.xin.6|li.xin|li.xin.7|li.xin.3|li.xin.4|li.xin.13|li.xin.2|li.xin.9"Wang, Shu"https://zbmath.org/authors/?q=ai:wang.shuSummary: This paper is concerned with smooth solutions of the non-isentropic Euler-Poisson system for ion dynamics. The system arises in the modeling of semi-conductor, in which appear one small parameter, the momentum relaxation time. When the initial data are near constant equilibrium states, with the help of uniform energy estimates and compactness arguments, we rigorously prove the convergence of the system for all time, as the relaxation time goes to zero. The limit system is the drift-diffusion system.Contractivity for Smoluchowski's coagulation equation with solvable kernelshttps://zbmath.org/1472.353812021-11-25T18:46:10.358925Z"Cañizo, José A."https://zbmath.org/authors/?q=ai:canizo.jose-alfredo"Lods, Bertrand"https://zbmath.org/authors/?q=ai:lods.bertrand"Throm, Sebastian"https://zbmath.org/authors/?q=ai:throm.sebastianSummary: We show that the Smoluchowski coagulation equation with the solvable kernels \(K(x,y)\) equal to 2, \(x+y\) or \(xy\) is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self-similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann-type equation, and extend already existing results on exponential convergence to self-similarity for Smoluchowski's coagulation equation.PDE/statistical mechanics duality: relation between Guerra's interpolated \(p\)-spin ferromagnets and the Burgers hierarchyhttps://zbmath.org/1472.353822021-11-25T18:46:10.358925Z"Fachechi, Alberto"https://zbmath.org/authors/?q=ai:fachechi.albertoSummary: We examine the duality relating the equilibrium dynamics of the mean-field \(p\)-spin ferromagnets at finite size in the Guerra's interpolation scheme and the Burgers hierarchy. In particular, we prove that -- for fixed \(p\) -- the expectation value of the order parameter on the first side w.r.t. the generalized partition function satisfies the \(p-1\)-th element in the aforementioned class of nonlinear equations. In the light of this duality, we interpret the phase transitions in the thermodynamic limit of the statistical mechanics model with the development of shock waves in the PDE side. We also obtain the solutions for the \(p\)-spin ferromagnets at fixed \(N\), allowing us to easily generate specific solutions of the corresponding equation in the Burgers hierarchy. Finally, we obtain an effective description of the finite \(N\) equilibrium dynamics of the \(p=2\) model with some standard tools in PDE side.Mathematical modeling of a temperature-sensitive and tissue-mimicking gel matrix: solving the Flory-Huggins equation for an elastic ternary mixture systemhttps://zbmath.org/1472.353832021-11-25T18:46:10.358925Z"Sung, Baeckkyoung"https://zbmath.org/authors/?q=ai:sung.baeckkyoungSummary: Programmed to retain active responsivity to environmental stimuli, diverse types of synthetic gels have been attracting interests regarding various applications, such as elastomer biodevices. In a different approach, when the gels are made of tissue-derived biopolymers, they can act as an artificial extracellular matrix (ECM) for use as soft implants in medicine. To explore the physical properties of hydrogels in terms of statistical thermodynamics, the mean-field Flory-Huggins-Rehner theory has long been used with various analytical and numerical modifications. Here, we suggest a novel mathematical model on the phase transition of a biological hybrid gel that is sensitive to ambient temperature. To mimic acellular soft tissues, the ECM-like hydrogel is modeled as a network of biopolymers, such as type I collagen and gelatin, which are covalently crosslinked and swollen in aqueous solvents. Within the network, thermoresponsive synthetic polymer chains are doped by chemical conjugation. Based on the Flory-Huggins-Rehner framework, our analytical model phenomenologically illustrates a well-characterized volume phase behavior of engineered tissue mimics as a function of temperature by formulating the ternary mixing free energy of the polymer-solvent system and by generalizing the elastic free energy term. With this formalism, the decoupling of the Flory-Huggins interaction parameter between the thermoresponsive polymer and ECM biopolymer enables deriving a simple steady-state formula for the volume phase transition as a function of the structural and compositional parameters. We show that the doping ratio of thermoresponsive polymers and the Flory-Huggins interaction parameter between biopolymer and water affect the phase transition temperature of the ECM-like gels.Periodic particle arrangements using standing acoustic waveshttps://zbmath.org/1472.353842021-11-25T18:46:10.358925Z"Vasquez, Fernando Guevara"https://zbmath.org/authors/?q=ai:guevara-vasquez.fernando"Mauck, China"https://zbmath.org/authors/?q=ai:mauck.chinaSummary: We determine crystal-like materials that can be fabricated by using a standing acoustic wave to arrange small particles in a non-viscous liquid resin, which is cured afterwards to keep the particles in the desired locations. For identical spherical particles with the same physical properties and small compared to the wavelength, the locations where the particles are trapped correspond to the minima of an acoustic radiation potential which describes the net forces that a particle is subject to. We show that the global minima of spatially periodic acoustic radiation potentials can be predicted by the eigenspace of a small real symmetric matrix corresponding to its smallest eigenvalue. We relate symmetries of this eigenspace to particle arrangements composed of points, lines or planes. Since waves are used to generate the particle arrangements, the arrangement's periodicity is limited to certain Bravais lattice classes that we enumerate in two and three dimensions.Homogenization and diffusion approximation of the Vlasov-Poisson-Fokker-Planck system: a relative entropy approachhttps://zbmath.org/1472.353852021-11-25T18:46:10.358925Z"Addala, Lanoir"https://zbmath.org/authors/?q=ai:addala.lanoir"El Ghani, Najoua"https://zbmath.org/authors/?q=ai:el-ghani.najoua"Tayeb, Mohamed Lazhar"https://zbmath.org/authors/?q=ai:tayeb.mohamed-lazharSummary: We are concerned with the analysis of the approximation by diffusion and homogenization of a Vlasov-Poisson-Fokker-Planck system. Here we generalize the convergence result of the second author and \textit{N. Masmoudi} [Commun. Math. Sci. 8, No. 2, 463--479 (2010; Zbl 1193.35228)] where the same problem is treated without the oscillating electrostatic potential and we extend the one dimensional result of the third author [Ann. Henri Poincaré 17, No. 9, 2529--2553 (2016; Zbl 1456.82798)] to the case of several space dimensions. An averaging lemma and two scale convergence techniques are used to prove rigorously the convergence of the scaled Vlasov-Poisson-Fokker-Planck system to a homogenized Drift-Diffusion-Poisson system.Spectrum analysis for the Vlasov-Poisson-Boltzmann systemhttps://zbmath.org/1472.353862021-11-25T18:46:10.358925Z"Li, Hai-Liang"https://zbmath.org/authors/?q=ai:li.hailiang"Yang, Tong"https://zbmath.org/authors/?q=ai:yang.tong"Zhong, Mingying"https://zbmath.org/authors/?q=ai:zhong.mingyingSummary: By identifying a norm capturing the effect of the forcing governed by the Poisson equation, we give a detailed spectrum analysis on the linearized Vlasov-Poisson-Boltzmann system around a global Maxwellian. It is shown that the electric field governed by the self-consistent Poisson equation plays a key role in the analysis so that the spectrum structure is genuinely different from the well-known one of the Boltzmann equation. Based on this, we give the optimal time decay rates of solutions to the equilibrium.Asymptotic stability of the phase-homogeneous solution to the Kuramoto-Sakaguchi equation with inertiahttps://zbmath.org/1472.353872021-11-25T18:46:10.358925Z"Choi, Young-Pil"https://zbmath.org/authors/?q=ai:choi.young-pil"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Xiao, Qinghua"https://zbmath.org/authors/?q=ai:xiao.qinghua"Zhang, Yinglong"https://zbmath.org/authors/?q=ai:zhang.yinglongFokker-Plank system for movement of micro-organism population in confined environmenthttps://zbmath.org/1472.353882021-11-25T18:46:10.358925Z"Fu, Jingyi"https://zbmath.org/authors/?q=ai:fu.jingyi"Perthame, Benoit"https://zbmath.org/authors/?q=ai:perthame.benoit"Tang, Min"https://zbmath.org/authors/?q=ai:tang.minSummary: We consider self-propelled particles confined between two parallel plates, moving with a constant velocity while their moving direction changes by rotational diffusion. The probability distribution of such micro-organisms in confined environment is singular because particles accumulate at the boundaries. This leads us to distinguish between the probability distribution densities in the bulk and in the boundaries. They satisfy a degenerate Fokker-Planck system and we propose boundary conditions that take into account the switching between free-moving and boundary-contacting particles. Relative entropy property, a priori estimates and the convergence to an unique steady state are established. The steady states of both the PDE and individual based stochastic models are compared numerically.Strong Feller property of the magnetohydrodynamics system forced by space-time white noisehttps://zbmath.org/1472.353892021-11-25T18:46:10.358925Z"Yamazaki, Kazuo"https://zbmath.org/authors/?q=ai:yamazaki.kazuoQuasigeostrophic equations for fractional powers of infinitesimal generatorshttps://zbmath.org/1472.353902021-11-25T18:46:10.358925Z"Abadias, Luciano"https://zbmath.org/authors/?q=ai:abadias.luciano"Miana, Pedro J."https://zbmath.org/authors/?q=ai:miana.pedro-jSummary: In this paper we treat the following partial differential equation, the quasigeostrophic equation: \(\left(\partial /\partial t + u \cdot \nabla\right) f = - \sigma \left(- A\right)^\alpha f, 0 \leq \alpha \leq 1\), where \((A, D(A))\) is the infinitesimal generator of a convolution \(C_0\)-semigroup of positive kernel on \(L^p(\mathbb{R}^n),\) with \(1 \leq p < \infty\). Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers \(- A)^\alpha\) for \(0 \leq \alpha \leq 1\). We use these estimates to obtain \(L^p\)-decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (taking \(A = \Delta\), the Laplacian), which has been studied in the literature.The stability principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinateshttps://zbmath.org/1472.353912021-11-25T18:46:10.358925Z"Cullen, M. J. P."https://zbmath.org/authors/?q=ai:cullen.michael-john-priestley"Kuna, T."https://zbmath.org/authors/?q=ai:kuna.tobias"Pelloni, B."https://zbmath.org/authors/?q=ai:pelloni.beatrice"Wilkinson, M."https://zbmath.org/authors/?q=ai:wilkinson.malcolm-h|wilkinson.mark-a|wilkinson.mark-d|wilkinson.michael-h-fSummary: The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimization argument originally inspired by the Stability Principle as studied by Cullen, Purser and others, uses optimal transport techniques as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of the Stability Principle in a rigorous mathematical framework.Invariant measures and global well posedness for the SQG equationhttps://zbmath.org/1472.353922021-11-25T18:46:10.358925Z"Földes, Juraj"https://zbmath.org/authors/?q=ai:foldes.juraj"Sy, Mouhamadou"https://zbmath.org/authors/?q=ai:sy.mouhamadouSummary: We construct an invariant measure \(\mu\) for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of \(\mu\) are initial conditions of global, unique solutions of SQG that depend continuously on the initial data. In addition, we show that the support of \(\mu\) is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measure a \(\mu\) is obtained via fluctuation-dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.Local well-posedness of strong solutions to the three-dimensional compressible primitive equationshttps://zbmath.org/1472.353932021-11-25T18:46:10.358925Z"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.4|liu.xin.3|liu.xin.2|liu.xin|liu.xin.1|liu.xin.5"Titi, Edriss S."https://zbmath.org/authors/?q=ai:titi.edriss-salehSummary: This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity.Monotone solutions for mean field games master equations: finite state space and optimal stoppinghttps://zbmath.org/1472.353942021-11-25T18:46:10.358925Z"Bertucci, Charles"https://zbmath.org/authors/?q=ai:bertucci.charlesSummary: We present a new notion of solution for mean field games master equations. This notion allows us to work with solutions which are merely continuous. We first prove results of uniqueness and stability for such solutions. It turns out that this notion is helpful to characterize the value function of mean field games of optimal stopping or impulse control and this is the topic of the second half of this paper. The notion of solution we introduce is only useful in the monotone case. In this article we focus on the finite state space case.Convergence and stochastic homogenization of a class of two components nonlinear reaction-diffusion systemshttps://zbmath.org/1472.353952021-11-25T18:46:10.358925Z"Anza, Hafsa Omar"https://zbmath.org/authors/?q=ai:anza.hafsa-omar"Mandallena, Jean Philippe"https://zbmath.org/authors/?q=ai:mandallena.jean-philippe"Michaille, Gérard"https://zbmath.org/authors/?q=ai:michaille.gerardSummary: We establish a convergence theorem for a class of two components nonlinear reaction-diffusion systems. Each diffusion term is the subdifferential of a convex functional of the calculus of variations whose class is equipped with the Mosco-convergence. The reaction terms are structured in such a way that the systems admit bounded solutions, which are positive in the modeling of ecosystems. As a consequence, under a stochastic homogenization framework, we prove two homogenization theorems for this class. We illustrate the results with the stochastic homogenization of a prey-predator model with saturation effect.The Strassen invariance principle for certain non-stationary Markov-Feller chainshttps://zbmath.org/1472.353962021-11-25T18:46:10.358925Z"Czapla, Dawid"https://zbmath.org/authors/?q=ai:czapla.dawid"Horbacz, Katarzyna"https://zbmath.org/authors/?q=ai:horbacz.katarzyna"Wojewódka-Ściążko, Hanna"https://zbmath.org/authors/?q=ai:wojewodka-sciazko.hannaSummary: We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov-Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression.Corrigendum to: ``A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices''https://zbmath.org/1472.353972021-11-25T18:46:10.358925Z"Ei, Shin-Ichiro"https://zbmath.org/authors/?q=ai:ei.shin-ichiro"Ishii, Hiroshi"https://zbmath.org/authors/?q=ai:ishii.hiroshi"Sato, Makoto"https://zbmath.org/authors/?q=ai:sato.makoto"Tanaka, Yoshitaro"https://zbmath.org/authors/?q=ai:tanaka.yoshitaro"Wang, Miaoxing"https://zbmath.org/authors/?q=ai:wang.miaoxing"Yasugi, Tetsuo"https://zbmath.org/authors/?q=ai:yasugi.tetsuoFig. 1b in the authors' paper [ibid. 81, No. 4--5, 981--1028 (2020; Zbl 1451.35234)] is corrected.Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomyhttps://zbmath.org/1472.353982021-11-25T18:46:10.358925Z"El-Hachem, Maud"https://zbmath.org/authors/?q=ai:el-hachem.maud"Mccue, Scott W."https://zbmath.org/authors/?q=ai:mccue.scott-william"Jin, Wang"https://zbmath.org/authors/?q=ai:jin.wang"Du, Yihong"https://zbmath.org/authors/?q=ai:du.yihong"Simpson, Matthew J."https://zbmath.org/authors/?q=ai:simpson.matthew-jSummary: The Fisher-Kolmogorov-Petrovsky-Piskunov model, also known as the Fisher-KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher-Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher-Stefan model is that it is able to simulate population extinction, giving rise to a \textit{spreading-extinction dichotomy}. In this work, we revisit travelling wave solutions of the Fisher-KPP model and show that these results provide new insight into travelling wave solutions of the Fisher-Stefan model and the spreading-extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher-Stefan model and often-neglected travelling wave solutions of the Fisher-KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher-Stefan model in the limit of slow travelling wave speeds, \(c \ll 1\).The Picone identity: a device to get optimal uniqueness results and global dynamics in population dynamicshttps://zbmath.org/1472.353992021-11-25T18:46:10.358925Z"Fernández-Rincón, Sergio"https://zbmath.org/authors/?q=ai:fernandez-rincon.sergio"López-Gómez, Julián"https://zbmath.org/authors/?q=ai:lopez-gomez.julianSummary: This paper infers from a generalized Picone identity the uniqueness of the stable positive solution for a class of semilinear equations of superlinear indefinite type, as well as the uniqueness and global attractivity of the coexistence state in two generalized diffusive prototypes of the symbiotic and competing species models of Lotka-Volterra. The optimality of these uniqueness theorems reveals the tremendous strength of the Picone identity.Analysis of spherical shell solutions for the radially symmetric aggregation equationhttps://zbmath.org/1472.354002021-11-25T18:46:10.358925Z"Guardia, Daniel Balagué"https://zbmath.org/authors/?q=ai:guardia.daniel-balague"Barbaro, Alethea"https://zbmath.org/authors/?q=ai:barbaro.alethea-b-t"Carrillo, Jose A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonio"Volkin, Robert"https://zbmath.org/authors/?q=ai:volkin.robertThis works studies the following aggregation equation for a density of particles \(\rho (t, x)\) with velocity \(v(t, x)\): \[\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho v)=0, \qquad v=-\nabla W * \rho. \] The considered repulsive-attractive potentials \(W\) are radially symmetric and of power type, and the analyzed solutions are radial probability distributions that contain spherical shells.
If \(d\) denotes the dimension of the space, a shell corresponds to a uniform distribution of particles on the surface of a \((d-1)\)-dimensional sphere, and, in radial coordinates, it is expressed as a Dirac delta function.
The techniques used in this paper rely on a pseudo-inverse formulation of the aggregation equation and on the study of the associated interaction energy functional. Some numerical simulations are also provided.Global existence and uniform boundedness in a chemotaxis model with signal-dependent motilityhttps://zbmath.org/1472.354012021-11-25T18:46:10.358925Z"Jiang, Jie"https://zbmath.org/authors/?q=ai:jiang.jie"Laurençot, Philippe"https://zbmath.org/authors/?q=ai:laurencot.philippeThe system \[u_t=\Delta(\gamma(v)u),\] \[0=\Delta v-v+u,\] is studied in bounded domains of \({\mathbb R}^N\), with the homogeneous Neumann boundary conditions for \(u\) and \(v\). Stationary solutions to that system solve the chemotaxis model \[u_t=\nabla\cdot(\nabla u-u\nabla\phi(v)),\] \[-\Delta v+v=u\] provided \(\gamma(v)=e^{-\phi(v)}\), whose solutions may blow up in some cases and thus, it is interesting to compare dynamical behaviors of solutions to both systems. Global-in-time classical and nonnegative solutions to the former system are shown to exist under a simple assumption: \(\gamma\in C^3((0,\infty))\), \(\gamma>0\), \(\sup_{\tau\in[s,\infty]}\gamma(\tau)<\infty\) for all \(s>0\). Note that sign changes of \(\gamma'\) are permitted by this assumption so that both attraction and repulsion effects can be present in that model. Under some addititional assumptions these solutions are uniformly bounded. This result improves some previous ones, and its proof involves new delicate iteration and monotonicity techniques.Singular limits of sign-changing weighted eigenproblemshttps://zbmath.org/1472.354022021-11-25T18:46:10.358925Z"Kielty, Derek"https://zbmath.org/authors/?q=ai:kielty.derekSummary: Consider the eigenvalue problem generated by a fixed differential operator with a sign-changing weight on the eigenvalue term. We prove that as part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. Our main theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. We apply our results to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type boundary conditions, and a problem where the eigenvalue appears in both the equation and the boundary condition.Large-time behavior of matured population in an age-structured modelhttps://zbmath.org/1472.354032021-11-25T18:46:10.358925Z"Li, Linlin"https://zbmath.org/authors/?q=ai:li.linlin"Ainseba, Bedreddine"https://zbmath.org/authors/?q=ai:ainseba.bedreddineSummary: In this paper, we model a mosquito plasticity problem and investigate the large time behavior of matured population under different control strategies. We prove that when the control is small, then the matured population will become large for large time and when the control is large, then the matured population will become small for large time. In the intermediate case, we derive a time-delayed model for the matured population which can be governed by a sub-equation and a super-equation. We prove the existence of traveling fronts for the sub-equation and use it to prove that the matured population will finally be between the positive states of the sub-equation and super-equation. At last, we present numerical simulations.Toward understanding the boundary propagation speeds in tumor growth modelshttps://zbmath.org/1472.354042021-11-25T18:46:10.358925Z"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guo"Tang, Min"https://zbmath.org/authors/?q=ai:tang.min"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.6"Zhou, Zhennan"https://zbmath.org/authors/?q=ai:zhou.zhennanGlobal Hopf bifurcation in a phytoplankton-zooplankton system with delay and diffusionhttps://zbmath.org/1472.354052021-11-25T18:46:10.358925Z"Liu, Ming"https://zbmath.org/authors/?q=ai:liu.ming.3|liu.ming.4|liu.ming|liu.ming.2|liu.ming.1"Cao, Jun"https://zbmath.org/authors/?q=ai:cao.jun"Xu, Xiaofeng"https://zbmath.org/authors/?q=ai:xu.xiaofengLong time behavior and stable patterns in high-dimensional polarity models of asymmetric cell divisionhttps://zbmath.org/1472.354062021-11-25T18:46:10.358925Z"Morita, Yoshihisa"https://zbmath.org/authors/?q=ai:morita.yoshihisa"Seirin-Lee, Sungrim"https://zbmath.org/authors/?q=ai:lee.sungrim-seirinSummary: Asymmetric cell division is one of the fundamental processes to create cell diversity in the early stage of embryonic development. During this process, the polarity formation in the cell membrane has been considered as a key process by which the entire polarity formation in the cytosol is controlled, and it has been extensively studied in both experiments and mathematical models. Nonetheless, a mathematically rigorous analysis of the polarity formation in the asymmetric cell division has been little explored, particularly for bulk-surface models. In this article, we deal with polarity models proposed for describing the PAR polarity formation in the asymmetric cell division of a \textit{C. elegans} embryo. Using a simpler but mathematically consistent model, we exhibit the long time behavior of the polarity formation of a bulk-surface cell. Moreover, we mathematically prove the existence of stable polarity solutions of the model equation in an arbitrary high-dimensional domain and analyse how the boundary position of polarity domain is determined. Our results propose that the existence and dynamics of the polarity in the asymmetric cell division can be understood universally in terms of basic mathematical structures.Stable steady-state solutions of some biological aggregation modelshttps://zbmath.org/1472.354072021-11-25T18:46:10.358925Z"Potts, Jonathan R."https://zbmath.org/authors/?q=ai:potts.jonathan-r"Painter, Kevin J."https://zbmath.org/authors/?q=ai:painter.kevin-jLower bounds of finite-time blow-up of solutions to a two-species Keller-Segel chemotaxis modelhttps://zbmath.org/1472.354082021-11-25T18:46:10.358925Z"Sathishkumar, G."https://zbmath.org/authors/?q=ai:sathishkumar.govindharaju"Shangerganesh, L."https://zbmath.org/authors/?q=ai:shangerganesh.lingeshwaran"Karthikeyan, S."https://zbmath.org/authors/?q=ai:karthikeyan.shanmugasundaramSummary: In this paper, we investigate the blow-up phenomena of non-negative solutions of a two-species Keller-Segel chemotaxis model with Lotka-Volterra competitive source terms. We estimate the lower bounds for the blow-up time of solutions of the model under the Neumann boundary conditions in a bounded domain \(\Omega\subset\mathbb {R}^n\), \(n\ge 1\). The first-order differential inequality technique is applied to determine the results in various space dimensions by using different auxiliary functions.Modeling the voltage distribution in a non-locally but globally electroneutral confined electrolyte medium: applications for nanophysiologyhttps://zbmath.org/1472.354092021-11-25T18:46:10.358925Z"Tricot, A."https://zbmath.org/authors/?q=ai:tricot.a"Sokolov, I. M."https://zbmath.org/authors/?q=ai:sokolov.igor-mikhailovich"Holcman, D."https://zbmath.org/authors/?q=ai:holcman.davidSummary: The distribution of voltage in sub-micron cellular domains remains poorly understood. In neurons, the voltage results from the difference in ionic concentrations which are continuously maintained by pumps and exchangers. However, it not clear how electro-neutrality could be maintained by an excess of fast moving positive ions that should be counter balanced by slow diffusing negatively charged proteins. Using the theory of electro-diffusion, we study here the voltage distribution in a generic domain, which consists of two concentric disks (resp. ball) in two (resp. three) dimensions, where a negative charge is fixed in the inner domain. When global but not local electro-neutrality is maintained, we solve the Poisson-Nernst-Planck equation both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and found that the voltage changes considerably on a spatial scale which is much larger than the Debye screening length, which assumes electro-neutrality. The present result suggests that long-range voltage drop changes are expected in neuronal microcompartments, probably relevant to explain the activation of far away voltage-gated channels located on the surface membrane.Influence of temperature on Turing pattern formationhttps://zbmath.org/1472.354102021-11-25T18:46:10.358925Z"van Gorder, Robert A."https://zbmath.org/authors/?q=ai:van-gorder.robert-aSummary: The Turing instability is one of the most commonly studied mechanisms leading to pattern formation in reaction-diffusion systems, yet there are still many open questions on the applicability of the Turing mechanism. Although experiments on pattern formation using chemical systems have shown that temperature differences play a role in pattern formation, there is far less theoretical work concerning the interplay between temperature and spatial instabilities. We consider a thermodynamically extended reaction-diffusion system, consisting of a pair of reaction-diffusion equations coupled to an energy equation for temperature, and use this to obtain a natural extension of the Turing instability accounting for temperature. We show that thermal contributions can restrict or enlarge the set of unstable modes possible under the instability, and in some cases may be used to completely shift the set of unstable modes, strongly modifying emergent Turing patterns. Spatial heterogeneity plays a role under several experimentally feasible configurations, and we give particular consideration to scenarios involving thermal gradients, thermodynamics of chemicals transported within a flow, and thermodiffusion. Control of Turing patterns is also an area of active interest, and we also demonstrate how patterns can be modified using time-dependent control of the boundary temperature.Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rateshttps://zbmath.org/1472.354112021-11-25T18:46:10.358925Z"Wang, Jinliang"https://zbmath.org/authors/?q=ai:wang.jinliang.1|wang.jinliang.2|wang.jinliang"Cui, Renhao"https://zbmath.org/authors/?q=ai:cui.renhaoSummary: This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment. We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.Traveling wave solution in a diffusive predator-prey system with Holling type-IV functional responsehttps://zbmath.org/1472.354122021-11-25T18:46:10.358925Z"Yang, Deniu"https://zbmath.org/authors/?q=ai:yang.deniu"Liu, Lihan"https://zbmath.org/authors/?q=ai:liu.lihan"Wang, Hongyong"https://zbmath.org/authors/?q=ai:wang.hongyongSummary: We establish the existence of traveling wave solution for a reaction-diffusion predator-prey system with Holling type-IV functional response. For simplicity, only one space dimension will be involved, the traveling solution equivalent to the heteroclinic orbits in \(R^3\). The methods used to prove the result are the shooting argument and the invariant manifold theory.When do Keller-Segel systems with heterogeneous logistic sources admit generalized solutions?https://zbmath.org/1472.354132021-11-25T18:46:10.358925Z"Yan, Jianlu"https://zbmath.org/authors/?q=ai:yan.jianlu"Fuest, Mario"https://zbmath.org/authors/?q=ai:fuest.marioA doubly parabolic chemotaxis system with space dependent logistic type sources is considered on bounded domains of \(\mathbb R^n\). Refined conditions for the global-in-time existence of suitable weak solutions have been derived improving previous results. A recourse to generalized solutions had been important in view of poor regularity properties in general cases.Corrigendum to: ``Analysis of a tumor-model free boundary problem with a nonlinear boundary condition''https://zbmath.org/1472.354142021-11-25T18:46:10.358925Z"Zheng, Jiayue"https://zbmath.org/authors/?q=ai:zheng.jiayue"Cui, Shangbin"https://zbmath.org/authors/?q=ai:cui.shangbinSome gaps in the proofs of Theorems 1.1 and 1.2 of the authors' paper [ibid. 478, No. 2, 806--824 (2019; Zbl 1421.35393)] are filled.Fine metrizable convex relaxations of parabolic optimal control problemshttps://zbmath.org/1472.354152021-11-25T18:46:10.358925Z"Roubíček, Tomáš"https://zbmath.org/authors/?q=ai:roubicek.tomasThe paper deals with fine metrizable convex relaxations of parabolic optimal control problems. In particular, a compromising convex compactification is devised. The basic idea consists in combining classical techniques for Young measures with Choquet theory. Therefore, the proposed approach works under classical \(\sigma\)-additive measures and standard sequences. At the same time, it allows for dealing with a wider class of nonlinearities than only affine. The controls \(u\) are valued in the set \(S_p\) of the form \[ S_p = \{ u \in L^p(\Omega;\mathbb{R}^m): u(x) \in B \; \text{for a.a.} \; x \in \Omega\}, \] with \(\Omega \subset \mathbb{R}^d\), \(d \in \mathbb{N}\), \(1 \leq p < +\infty\) and \(B \subset \mathbb{R}^m\) bounded and closed. In addition, some generalization to unbounded domain \(B\) by considering a general \(S_p\) bounded in \(L^p(\Omega;\mathbb{R}^m)\), with \(1 \leq p < \infty\) fixed but not necessarily bounded in \(L^\infty(\Omega;\mathbb{R}^m)\), is also discussed. Finally, an application to optimal control of a system of semilinear parabolic differential equations is presented for the reader convenience, together with other relaxation strategies as well as more general nonlinearities, showing that the finds reported in the paper are useful for practical applications.A priori estimates for semistable solutions of \(p\)-Laplace equations with general nonlinearityhttps://zbmath.org/1472.354162021-11-25T18:46:10.358925Z"Aghajani, A."https://zbmath.org/authors/?q=ai:aghajani.aydin|aghajani.asadollah|aghajani.a-h"Mottaghi, S. F."https://zbmath.org/authors/?q=ai:mottaghi.s-fSummary: In this paper we consider the \(p\)-Laplace equation \(- \Delta_pu=\lambda f(u)\) in a smooth bounded domain \(\Omega\subset \mathbb{R}^N\) with zero Dirichlet boundary condition, where \(p>1\), \(\lambda>0\) and \(f:[0,\infty)\to\mathbb{R}\) is a \(C^1\) function with \(f(0)>0\), \(f'\geqslant 0\) and \(\lim_{t\to\infty}\frac{f(t)}{t^p-1}=\infty\). For the sequence \((u_\lambda)_{0<\lambda<\lambda^\ast}\) of minimal semi-stable solutions, by applying the semi-stability inequality we find a class of functions \(E\) that asymptotically behave like a power of \(f\) at infinity and show that \(\|E(u_\lambda)\|_{L^1(\Omega)}\) is uniformly bounded for \(\lambda<\lambda^\ast\). Then using elliptic regularity theory we provide some new \(L^\infty\) estimates for the extremal solution \(u^\ast\), under some suitable conditions on the nonlinearity \(f\), where the obtained results require neither the convexity of \(f\) nor the strictly convexity of the domain. In particular, under some mild assumptions on \(f\) we show that \(u^\ast\in L^\infty(\Omega)\) for \(N<p+4p/(p-1)\), which is conjectured to be the optimal regularity dimension for \(u^\ast\).The heat equation on the finite Poincaré upper half-planehttps://zbmath.org/1472.354172021-11-25T18:46:10.358925Z"Dedeo, M. R."https://zbmath.org/authors/?q=ai:dedeo.michelle-r"Velasquez, Elinor"https://zbmath.org/authors/?q=ai:velasquez.elinorSummary: A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting.Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphshttps://zbmath.org/1472.354182021-11-25T18:46:10.358925Z"Hofmann, Matthias"https://zbmath.org/authors/?q=ai:hofmann.matthias"Kennedy, James B."https://zbmath.org/authors/?q=ai:kennedy.james-bSummary: We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [the second author et al., Calc. Var. Partial Differ. Equ. 60, No. 2, Paper No. 61, 63 p. (2021; Zbl 1462.35222)]. These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of \textit{cutting} a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its \textit{rank}. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander's inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.A heat flow for the mean field equation on a finite graphhttps://zbmath.org/1472.354192021-11-25T18:46:10.358925Z"Lin, Yong"https://zbmath.org/authors/?q=ai:lin.yong"Yang, Yunyan"https://zbmath.org/authors/?q=ai:yang.yunyanSummary: Inspired by works of \textit{J.-B. Castéras} [Pac. J. Math. 276, No. 2, 321--345 (2015; Zbl 1331.53097)], \textit{J. Li} and \textit{C. Zhu} [Calc. Var. Partial Differ. Equ. 58, No. 2, Paper No. 60, 18 p. (2019; Zbl 1415.58014)], \textit{L. Sun} and \textit{J. Zhu} [Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 42, 26 p. (2021; Zbl 1458.35437)], we propose a heat flow for the mean field equation on a connected finite graph \(G=(V,E)\). Namely
\[
\begin{cases}
\partial_t\phi (u)=\Delta u-Q+\rho \frac{e^u}{\int_V e^u d\mu}\\
u(\cdot ,0)=u_0,
\end{cases}
\]
where \(\Delta\) is the standard graph Laplacian, \(\rho\) is a real number, \(Q:V\rightarrow\mathbb{R}\) is a function satisfying \(\int_VQd\mu =\rho\), and \(\phi :\mathbb{R}\rightarrow\mathbb{R}\) is one of certain smooth functions including \(\phi (s)=e^s\). We prove that for any initial data \(u_0\) and any \(\rho \in\mathbb{R} \), there exists a unique solution \(u:V\times [0,+\infty)\rightarrow\mathbb{R}\) of the above heat flow; moreover, \(u(x, t)\) converges to some function \(u_\infty :V\rightarrow\mathbb{R}\) uniformly in \(x\in V\) as \(t\rightarrow +\infty \), and \(u_\infty\) is a solution of the mean field equation
\[
\Delta u_{\infty}-Q+\rho \frac{e^{u_\infty}}{\int_V e^{u_\infty} d\mu}=0.
\]
Though \(G\) is a finite graph, this result is still unexpected, even in the special case \(Q\equiv 0\). Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz-Simon type inequality and use it to conclude the convergence of the heat flow.On local Lipschitz regularity for quasilinear equations in the Heisenberg grouphttps://zbmath.org/1472.354202021-11-25T18:46:10.358925Z"Mukherjee, Shirsho"https://zbmath.org/authors/?q=ai:mukherjee.shirshoSummary: The goal of this article is to establish local Lipschitz continuity of solutions for a class of sub-elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity condition is a natural generalization of the sub-elliptic \(p\)-Laplace equation and more general quasilinear equations with polynomial or exponential type growth.Mild solutions are weak solutions in a class of (non)linear measure-valued evolution equations on a bounded domainhttps://zbmath.org/1472.354212021-11-25T18:46:10.358925Z"Evers, Joep H. M."https://zbmath.org/authors/?q=ai:evers.joep-h-mSummary: We study the connection between mild and weak solutions for a class of measure-valued evolution equations on the bounded domain \([0,1]\). Mass moves, driven by a velocity field that is either a function of the spatial variable only, \(v=v(x0\), or depends on the solution \(\mu\) itself: \(v=v[\mu](x)\). The flow is stopped at the boundaries of \([0,1]\), while mass is gated away by a certain right-hand side. In previous works \textit{J. H. M. Evers} et al. [J. Differ. Equations 259, No. 3, 1068--1097 (2015; Zbl 1315.35057); SIAM J. Math. Anal. 48, No. 3, 1929--1953 (2016; Zbl 1342.28004)], we showed the existence and uniqueness of appropriately defined mild solutions for \(v=v(x)\) and \(v=v[\mu](x0\), respectively. In the current paper we define weak solutions (by specifying the weak formulation and the space of test functions). The main result is that the aforementioned mild solutions are weak solutions, both when \(v=v(x)\) and when \(v=v[\mu](x)\).The Dirichlet problem for nonlocal elliptic equationshttps://zbmath.org/1472.354222021-11-25T18:46:10.358925Z"Tian, Rongrong"https://zbmath.org/authors/?q=ai:tian.rongrong"Wei, Jinlong"https://zbmath.org/authors/?q=ai:wei.jinlong"Tang, Yanbin"https://zbmath.org/authors/?q=ai:tang.yanbinSummary: We study a class of nonlocal elliptic equations \((-\Delta)^{\alpha/2}\rho(x)-b(x)\cdot\nabla\rho(x)=f(x)\) on a bounded domain \(\Omega\subset\mathbb{R}^d\). For \(f\in L^q(\Omega)(q>d/\alpha)\), by the Lax-Milgram theorem and the De Giorgi iteration, we prove the existence and uniqueness of \(L^\infty\) solution. Furthermore, we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations.Free boundary problem in a polymer solution modelhttps://zbmath.org/1472.354232021-11-25T18:46:10.358925Z"Petrova, A. G."https://zbmath.org/authors/?q=ai:petrova.anna-georgevna"Pukhnachev, V. V."https://zbmath.org/authors/?q=ai:pukhnachov.vladislav-vThe authors consider the integro-differential equation
\[
\frac{\partial w}{\partial t}+\frac{\partial w}{\partial y}\int_{0}^{y}w(z,t)dz-w^{2}=\frac{\partial^{2}w}{\partial y^{2}}+\gamma (\frac{\partial^{3}w}{\partial y^{2}\partial t}+\frac{\partial^{3}w}{\partial y^{3}}\int_{0}^{y}w(z,t)dz- \frac{\partial^{2}w}{\partial y^{2}}),
\]
posed in the domain \(\Omega_{T} = \{y,t:0 < y < h(t)\), \(0\leq t\leq T\}\). This model accounts for the flow of a mixture of water and polymer. This equation is completed with: \(\frac{dh}{dt}=\int_{0}^{h}w(y,t)dy\). The boundary conditions \(w(0,t)=0\) and \(\frac{\partial w}{\partial y}+\gamma (\frac{\partial^{2}w}{\partial y\partial t}+ \frac{\partial^{2}w}{\partial y^{2}}\int_{0}^{h}w(y,t)dy-w\frac{\partial w}{\partial y})(h(t),t)=0\) are added, together with the initial conditions \( w(y,0)=w_{0}(y)\), \(0\leq y\leq 1\), \(h(0)=1\). Here \(\gamma >0\) is a constant and \(w_{0}\) is a smooth (\(C^{3}\)) function of \(y\) satisfying the conditions \( w_{0}(0)=w_{0}^{\prime}(1)=0\). The first main result proves the existence of a local in time strong solution (\(h\in C^{1}([0,t^{\ast}])\), \(w\in C^{3,1}([0,h(t)]\times \lbrack 0,t^{\ast}])\)) to this problem. If the initial condition further satisfies \(w_{0}(y)\leq 0\), \(w_{0}(y)-\gamma w_{0}^{\prime \prime}(y)\leq 0\), the authors prove the existence of a classical solution \(h\in C^{1}([0,T])\), \(w\in C^{3,1}([0,h(t)]\times \lbrack 0,T])\) to the above problem. Both existence results are obtained through appropriate transformations and using Schauder's theorem. The authors then consider the case where \(\gamma\) tends to 0 and they observe that the problem turns into that of the deformation of a strip of viscous fluid. They here prove that the solution to this problem is destructed in finite time. They finally introduce asymptotic expansions with respect to \(\gamma\) and they express the second term of this asymptotic expansion.Discrete curve flows in two-dimensional Cayley-Klein geometrieshttps://zbmath.org/1472.354242021-11-25T18:46:10.358925Z"Benson, Joseph"https://zbmath.org/authors/?q=ai:benson.joseph"Valiquette, Francis"https://zbmath.org/authors/?q=ai:valiquette.francisSummary: Using the method of equivariant moving frames, we study geometric flows of discrete curves in the nine Cayley-Klein planes. We show that, under a certain arc-length preserving flow, the curvature invariant \(\kappa_ n\) evolves according to the differential-difference equation \(\frac{\partial \kappa_n}{\partial t} = (1+\epsilon \kappa_{n+1}^2)(\kappa_{n+1}-\kappa_{n-1})\), where the value of \(\varepsilon \in\{-1, 0, 1\}\) is linked to the geometry of the Cayley-Klein plane.
For the entire collection see [Zbl 1471.81009].On critical Kirchhoff problems driven by the fractional Laplacianhttps://zbmath.org/1472.354252021-11-25T18:46:10.358925Z"Appolloni, Luigi"https://zbmath.org/authors/?q=ai:appolloni.luigi"Molica Bisci, Giovanni"https://zbmath.org/authors/?q=ai:molica-bisci.giovanni"Secchi, Simone"https://zbmath.org/authors/?q=ai:secchi.simoneSummary: We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the Sobolev embedding sense. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.Traveling wave solutions for the space-time fractional \((2+1)\)-dimensional Calogero-Bogoyavlenskii-Schiff equation via two different methodshttps://zbmath.org/1472.354262021-11-25T18:46:10.358925Z"Bekhouche, Fares"https://zbmath.org/authors/?q=ai:bekhouche.fares"Komashynska, Iryna"https://zbmath.org/authors/?q=ai:komashynska.iryna-volodymyrivnaSummary: In this paper, we investigate the unified method and the modified Kudryashov method for obtaining exact traveling wave solutions of conformable fractional partial differential equations. In addition, a connection is given between the two methods. Then, using these methods, we obtain new exact solutions for the space-time fractional \((2+1)\)-dimensional Calogero-Bogoyavlensky-Schiff equation. Fractional derivatives are described in conformable sense. Various solutions have been obtained, including one-soliton, kink, anti-kink, periodic wave solutions, and multiple-soliton solutions. We also provide a graphical representation of some interesting exact solutions to the equation and discuss the behavior of these solutions. The considered methods can be effectively applied to a wide range of nonlinear fractional partial differential equations.Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacianhttps://zbmath.org/1472.354272021-11-25T18:46:10.358925Z"Bidi, Younes"https://zbmath.org/authors/?q=ai:bidi.younes"Beniani, Abderrahmane"https://zbmath.org/authors/?q=ai:beniani.abderrahmane"Zennir, Khaled"https://zbmath.org/authors/?q=ai:zennir.khaled"Himadan, Ahmed"https://zbmath.org/authors/?q=ai:himadan.ahmedSummary: We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.Local fractional Moisil-Teodorescu operator in quaternionic setting involving Cantor-type coordinate systemshttps://zbmath.org/1472.354282021-11-25T18:46:10.358925Z"Bory-Reyes, Juan"https://zbmath.org/authors/?q=ai:bory-reyes.juan"Pérez-de la Rosa, Marco Antonio"https://zbmath.org/authors/?q=ai:perez-de-la-rosa.marco-antonioSummary: The Moisil-Teodorescu operator is considered to be a good analogue of the usual Cauchy-Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in \(\mathbb{R}^3\). In the present work, a general quaternionic structure is developed for the local fractional Moisil-Teodorescu operator in Cantor-type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil-Teodorescu operator.Some new exact results for non-linear space-fractional diffusivity equationshttps://zbmath.org/1472.354292021-11-25T18:46:10.358925Z"Caserta, Arrigo"https://zbmath.org/authors/?q=ai:caserta.arrigo"Garra, Roberto"https://zbmath.org/authors/?q=ai:garra.roberto"Salusti, Ettore"https://zbmath.org/authors/?q=ai:salusti.ettoreSummary: In this paper we reconsider the classical nonlinear diffusivity equation of real gas in an heterogenous porous medium in light of the recent studies about nonlocal space-fractional generalizations of diffusion models. The obtained equation can be simply linearized into a classical space-fractional diffusion equation, widely studied in the literature. We consider the case of a power-law pressure-dependence of the permeability coefficient. In this case we provide some useful new exact analytical results. In particular, we are able to find a Barenblatt-type solution for a space-fractional Boussinesq equation, arising in this context.
For the entire collection see [Zbl 1470.26002].Fractional diffusive waves in the Cauchy and signalling problemshttps://zbmath.org/1472.354302021-11-25T18:46:10.358925Z"Consiglio, Armando"https://zbmath.org/authors/?q=ai:consiglio.armando"Mainardi, Francesco"https://zbmath.org/authors/?q=ai:mainardi.francescoSummary: This work deals with the results and the simulations obtained for the time-fractional diffusion-wave equation, i.e. a diffusion-like linear integro partial differential equation containing a pseudo-differential operator interpreted as a fractional derivative in time. The data function (initial signal) is provided by a box-function and the solutions are so obtained by a convolution of the Green function with the initial data function. The relevance of the topic lies in the possibility of describing physical processes that interpolates between the different responses of the diffusion and waves equations, equipped with a physically realistic initial signal. Here two problems are considered where the use of the Laplace transform in the analysis of the problems has lead since 1990s to special functions of the Wright type.
For the entire collection see [Zbl 1470.26002].Fractional-order model for cooling of a semi-infinite body by radiationhttps://zbmath.org/1472.354312021-11-25T18:46:10.358925Z"Esmaeili, Shahrokh"https://zbmath.org/authors/?q=ai:esmaeili.shahrokhSummary: In this paper, the fractional-order model for cooling of a semi-infinite body by radiation is considered.
In the supposed semi-infinite body, the equation of heat along with an initial condition and an asymptotic boundary condition form an equivalent equation in which the order of derivatives is halved.
This equation and a boundary condition introduced by the radiation heat transfer give rise to an initial value problem, whose differential equation is nonlinear and fractional order.
The semi-analytical solution to this nonlinear model was determined asymptotically at small and large times.
Moreover, two numerical methods including Grünwald-Letnikov approximation and Müntz-Legendre approximation yield numerical solutions to the problem.Some extension results for nonlocal operators and applicationshttps://zbmath.org/1472.354322021-11-25T18:46:10.358925Z"Ferrari, Fausto"https://zbmath.org/authors/?q=ai:ferrari.faustoSummary: In this paper, we deal with some recent and old results, concerning fractional operators, obtained via the extension technique. This approach is particularly fruitful for exploiting some of those well known properties, true for the local operators obtained via the extension approach, for deducing some parallel results about the underlaying nonlocal operators.
For the entire collection see [Zbl 1470.26002].Application of the fractional Sturm-Liouville theory to a fractional Sturm-Liouville telegraph equationhttps://zbmath.org/1472.354332021-11-25T18:46:10.358925Z"Ferreira, M."https://zbmath.org/authors/?q=ai:ferreira.michel|ferreira.max|ferreira.miguel-h|ferreira.marta|ferreira.m-n|ferreira.m-d-c|ferreira.mario-f-s|ferreira.maria-c-f|ferreira.maria-joao.2|ferreira.m-p|ferreira.maria-teodora|ferreira.mauricio-a|ferreira.miguel-jorge-bernabe|ferreira.maria-joao.1|ferreira.marco-s|ferreira.marina-a|ferreira.marcelo-rodrigo-portela|ferreira.marisa|ferreira.mauro-s|ferreira.marco-a-r|ferreira.marizete-a-c|ferreira.marcio-j-r|ferreira.mardson|ferreira.marcos-r-s|ferreira.manoel-m-jun|ferreira.mariana|ferreira.m-margarida-a|ferreira.m-luisa|ferreira.marcio-v|ferreira.milton|ferreira.maria-margarida|ferreira.marcelo-c"Rodrigues, M. M."https://zbmath.org/authors/?q=ai:rodrigues.maria-manuela|rodrigues.maikol-m"Vieira, N."https://zbmath.org/authors/?q=ai:vieira.newton-j|vieira.nelsonSummary: In this paper, we consider a non-homogeneous time-space-fractional telegraph equation in \(n\)-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm-Liouville operator defined in terms of right and left fractional Riemann-Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper some illustrative examples are presented.Existence of the gauge for fractional Laplacian Schrödinger operatorshttps://zbmath.org/1472.354342021-11-25T18:46:10.358925Z"Frazier, Michael W."https://zbmath.org/authors/?q=ai:frazier.michael-w"Verbitsky, Igor E."https://zbmath.org/authors/?q=ai:verbitsky.igor-eSummary: Let \(\Omega\subseteq\mathbb{R}^n\) be an open set, where \(n\geq 2\). Suppose \(\omega\) is a locally finite Borel measure on \(\Omega\). For \(\alpha\in (0,2)\), define the fractional Laplacian \((-\Delta)^{\alpha/2}\) via the Fourier transform on \(\mathbb{R}^n\), and let \(G\) be the corresponding Green's operator of order \(\alpha\) on \(\Omega\). Define \(T(u)=G(u\omega)\). If \(\Vert T\Vert_{L^2(\omega)\rightarrow L^2(\omega)}<1\), we obtain a representation for the unique weak solution \(u\) in the homogeneous Sobolev space \(L^{\alpha/2,2}_0(\Omega)\) of
\[
(-\Delta)^{\alpha/2} u=u\omega+\nu\text{ on }\Omega,\quad u=0\text{ on }\Omega^c,
\]
for \(\nu\) in the dual Sobolev space \(L^{-\alpha/2,2}(\Omega)\). If \(\Omega\) is a bounded \(C^{1,1}\) domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when \(\nu=\chi_{\Omega}\). These estimates are used to study the existence of a solution \(u_1\) (called the ``gauge'') of the integral equation \(u_1=1+G(u_1\omega)\) corresponding to the problem
\[
(-\Delta)^{\alpha/2}u=u\omega\text{ on }\Omega,\quad u\geq 0\text{ on }\Omega,\quad u=1\text{ on }\Omega^c.
\]
We show that if \(\Vert T\Vert<1\), then \(u_1\) always exists if \(0<\alpha<1\). For \(1\leq\alpha<2\), a solution exists if the norm of \(T\) is sufficiently small. We also show that the condition \(\Vert T\Vert <1\) does not imply the existence of a solution if \(1<\alpha<2\). The condition \(\Vert T\Vert\leq 1\) is necessary for the existence of \(u_1\) for all \(0<\alpha\leq 2\).Conformal boundary operators, \(T\)-curvatures, and conformal fractional Laplacians of odd orderhttps://zbmath.org/1472.354352021-11-25T18:46:10.358925Z"Gover, A. Rod"https://zbmath.org/authors/?q=ai:gover.ashwin-rod"Peterson, Lawrence J."https://zbmath.org/authors/?q=ai:peterson.lawrence-jSummary: We construct continuously parametrised families of conformally invariant boundary operators on densities. These generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with conformal Laplacian power operators, the boundary operators yield conformally invariant fractional Laplacian pseudodifferential operators on the boundary of a conformal manifold with boundary. We also find and construct new curvature quantities associated to our new operator families. These have links to the Branson \(Q\)-curvature and include higher-order generalisations of the mean curvature and the \(T\)-curvature of Chang-Qing. In the case of the standard conformal hemisphere, the boundary operator construction is particularly simple; the resulting operators provide an elementary construction of families of symmetry breaking intertwinors between the spherical principal series representations of the conformal group of the equator, as studied by Juhl and others. We discuss applications of our results and techniques in the setting of Poincaré-Einstein manifolds and also use our constructions to shed light on some conjectures of Juhl.On the wave solutions of time-fractional Sawada-Kotera-Ito equation arising in shallow waterhttps://zbmath.org/1472.354362021-11-25T18:46:10.358925Z"Jena, Rajarama Mohan"https://zbmath.org/authors/?q=ai:jena.rajarama-mohan"Chakraverty, Snehashish"https://zbmath.org/authors/?q=ai:chakraverty.snehashish"Jena, Subrat Kumar"https://zbmath.org/authors/?q=ai:jena.subrat-kumar"Sedighi, Hamid M."https://zbmath.org/authors/?q=ai:sedighi.hamid-mohammadSummary: A widescale description of various phenomena in science and engineering such as physics, chemical, acoustics, control theory, finance, economics, mechanical engineering, civil engineering, and social sciences is well described by nonlinear fractional differential equations (NLFDEs). In turbulence, fluid dynamics, and nonlinear biological systems, applications of NLFDEs can also be found. NLFDEs are believed to be powerful tools to describe real-world problems more precisely than the differential equation of the integer-order. In this research, we have used the fractional reduced differential transform method (FRDTM) to find the solution of the time-fractional Sawada-Kotera-Ito seventh-order equation. The novelty of the FRDTM is that it does not require any discretization, transformation, perturbation, or any restrictive conditions. In addition, compared to other methods, this approach needs less calculation. For special cases of an integer and noninteger orders, computed results are compared with existing results. Present results are in good agreement with the existing solutions. Here, the fractional derivatives are considered in the Caputo sense. Convergence analysis of the results has also been studied with the increasing number of terms of the solution.Final value problem for parabolic equation with fractional Laplacian and Kirchhoff's termhttps://zbmath.org/1472.354372021-11-25T18:46:10.358925Z"Luc, Nguyen Hoang"https://zbmath.org/authors/?q=ai:luc.nguyen-hoang"Kumar, Devendra"https://zbmath.org/authors/?q=ai:kumar.devendra.4|kumar.devendra.1|kumar.devendra.2|kumar.devendra|kumar.devendra.3"Long, Le Dinh"https://zbmath.org/authors/?q=ai:long.le-dinh"Van, Ho Thi Kim"https://zbmath.org/authors/?q=ai:van.ho-thi-kimSummary: In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In addition, to investigate the regularity, we also further study the nonwell-posed and give the regularized methods to get the correct approximate solution. With reasonable and appropriate input conditions, we can prove that the error between the regularized solution and the search solution is towards zero when \(\delta\) tends to zero.Existence of ground states of fractional Schrödinger equationshttps://zbmath.org/1472.354382021-11-25T18:46:10.358925Z"Ma, Li"https://zbmath.org/authors/?q=ai:ma.li.1|ma.li"Li, Zhenxiong"https://zbmath.org/authors/?q=ai:li.zhenxiongSummary: We consider ground states of the nonlinear fractional Schrödinger equation with potentials
\[
(-\Delta)^s u+V(x)u = f(x,u), \quad s\in(0,1),
\]
on the whole space \(\mathbb{R}^N\), where \(V\) is a periodic non-negative nontrivial function on \(\mathbb{R}^N\) and the nonlinear term \(f\) has some proper growth on \(u\). Under uniform bounded assumptions about \(V\), we can show the existence of a ground state. We extend the result of Li, Wang, and Zeng to the fractional case.A degenerate Kirchhoff-type inclusion problem with nonlocal operatorhttps://zbmath.org/1472.354392021-11-25T18:46:10.358925Z"Motreanu, Dumitru"https://zbmath.org/authors/?q=ai:motreanu.dumitruSummary: The chapter focuses on a Kirchhoff-type elliptic inclusion problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points and for which the multivalued term is in the form of the generalized gradient of a locally Lipschitz function. The corresponding elliptic equation has been treated in (Liu et al., Existence of solutions to Kirchhoff-type problem with vanishing nonlocal term and fractional \(p\)-Laplacian). Multiple nontrivial solutions are obtained by applying the nonsmooth critical point theory combined with truncation techniques.
For the entire collection see [Zbl 1470.49002].Existence and non-existence results for fractional Kirchhoff Laplacian problemshttps://zbmath.org/1472.354402021-11-25T18:46:10.358925Z"Nyamoradi, Nemat"https://zbmath.org/authors/?q=ai:nyamoradi.nemat"Ambrosio, Vincenzo"https://zbmath.org/authors/?q=ai:ambrosio.vincenzoSummary: In this paper, we study the following fractional Kirchhoff-type problem:
\[\begin{aligned}
\left[ a+b\Big (\iint_{{\mathbb{R}}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy \Big )^{\theta -1}\right] (-\Delta )^s u= &|u|^{2^*_s- 2} u\\
&\;+\lambda f(x) |u|^{q-2}u, \text{ in }\mathbb{R}^N,
\end{aligned}\]
where \((-\Delta )^s\) is the fractional Laplacian operator with \(0<s<1\), \(\lambda \ge 0\), \(a \ge 0\), \(b> 0\), \(1<q<2\), \(N>2s\), and \(2^*_s= \frac{2 N}{N-2s}\) is fractional critical Sobolev exponent. When \(\lambda =0\), under suitable values of the parameters \(\theta\), \(a\) and \(b\), we obtain a non-existence result and the existence of infinitely many nontrivial solutions for the above problem. Also, for suitable weight function \(f(x)\), using the Nehari manifold technique and the fibbing maps, we prove the existence of at least two positive solutions for a sufficiently small choice of \(\lambda\).Analysis of dual Bernstein operators in the solution of the fractional convection-diffusion equation arising in underground water pollutionhttps://zbmath.org/1472.354412021-11-25T18:46:10.358925Z"Sayevand, K."https://zbmath.org/authors/?q=ai:sayevand.khosro"Machado, J. Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiro"Masti, I."https://zbmath.org/authors/?q=ai:masti.imanSummary: The Bernstein operators (BO) are not orthogonal, but they have duals, which are obtained by a linear combination of BO. In recent years dual BO have been adopted in computer graphics, computer aided geometric design, and numerical analysis. This paper presents a numerical method based on the Bernstein operational matrices to solve the time-space fractional convection-diffusion equation. A generalization of the derivative matrix operator of fractional order and the error analysis are discussed. Numerical examples compare the proposed approach with previous works, showing that the method is more accurate and efficient.Compact embedding theorems and a Lions' type lemma for fractional Orlicz-Sobolev spaceshttps://zbmath.org/1472.354422021-11-25T18:46:10.358925Z"Silva, Edcarlos D."https://zbmath.org/authors/?q=ai:da-silva.edcarlos-domingos"Carvalho, M. L."https://zbmath.org/authors/?q=ai:carvalho.marcos-l-m|carvalho.marcos-leandro"de Albuquerque, J. C."https://zbmath.org/authors/?q=ai:de-albuquerque.jose-carlos"Bahrouni, Sabri"https://zbmath.org/authors/?q=ai:bahrouni.sabriSummary: In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' ``vanishing'' Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schrödinger equations, taking into account unbounded or bounded potentials.New idea of Atangana-Baleanu time-fractional derivative to advection-diffusion equationhttps://zbmath.org/1472.354432021-11-25T18:46:10.358925Z"Tlili, Iskander"https://zbmath.org/authors/?q=ai:tlili.iskander"Shah, Nehad Ali"https://zbmath.org/authors/?q=ai:shah.nehad-ali"Ullah, Saif"https://zbmath.org/authors/?q=ai:ullah.saif"Manzoor, Humera"https://zbmath.org/authors/?q=ai:manzoor.humeraSummary: The analytical study of one-dimensional generalized fractional advection-diffusion equation with a time-dependent concentration source on the boundary is carried out. The generalization consists into considering the advection-diffusion equation with memory based on the time-fractional Atangana-Baleanu derivative with Mittag-Leffler kernel. Analytical solution of the fractional differential advection-diffusion equation along with initial and boundary value conditions has been determined by employing Laplace transform and finite sine-Fourier transform. On the basis of the properties of Atangana-Baleanu fractional derivatives and the properties of Mittag-Leffler functions, the general solution is particularized for the fractional parameter \(\alpha = 1\) in order to find solution of the classical advection-diffusion process. The influence of memory parameter on the solute concentration has been investigated using the analytical solution and the software Mathcad. From this analysis, it is found that for a constant concentration's source on the boundary, the solute concentration is increasing with fractional parameter, and therefore, an advection-diffusion process described by Atangana-Baleanu time-fractional derivative leads to a smaller solute concentration than in the classical process.Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growthhttps://zbmath.org/1472.354442021-11-25T18:46:10.358925Z"Xiang, Mingqi"https://zbmath.org/authors/?q=ai:xiang.mingqi"Hu, Die"https://zbmath.org/authors/?q=ai:hu.die"Zhang, Binlin"https://zbmath.org/authors/?q=ai:zhang.binlin"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.6The paper is to prove the existence of two solutions for a class of degenerate Kirchhoff type problems driven by the fractional Laplace operator with variable order derivative and variable exponents, through using the fiber mapping approaches.Multiplicity of nonnegative solutions for a class of fractional p-Laplacian system in \(\mathbb{R}^N\)https://zbmath.org/1472.354452021-11-25T18:46:10.358925Z"Yaghoobi, Farajollah Mohammadi"https://zbmath.org/authors/?q=ai:yaghoobi.farajollah-mohammadi"Karamikabir, Nasrin"https://zbmath.org/authors/?q=ai:karamikabir.nasrinSummary: Here, we deal with the existence and multiplicity of nonnegative solutions for a class of fractional p-laplacian system in \(\mathbb{R}^N\). Using the variational methods and extracting Palais-Smale sequences in the Nehari manifold, we prove that there exists \(\lambda^\ast\) such that for \(\lambda\in(0,\lambda^\ast)\) the given problem has at least two distinct positive solutions.Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equationhttps://zbmath.org/1472.354462021-11-25T18:46:10.358925Z"Zhang, Zhi-Yong"https://zbmath.org/authors/?q=ai:zhang.zhiyongSummary: We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.Existence of three nontrivial solutions of asymptotically linear second order operator equationshttps://zbmath.org/1472.354472021-11-25T18:46:10.358925Z"Chen, Yingying"https://zbmath.org/authors/?q=ai:chen.yingyingSummary: In this article, we prove the existence of three nontrivial solutions for some second order operator equations, especially the asymptotically linear ones. The main methods are the Leray-Schauder degree theory and mountain pass theorem.Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equationshttps://zbmath.org/1472.354482021-11-25T18:46:10.358925Z"Bao, Ngoc Tran"https://zbmath.org/authors/?q=ai:bao.ngoc-tran"Hoang, Luc Nguyen"https://zbmath.org/authors/?q=ai:hoang.luc-nguyen"Van, Au Vo"https://zbmath.org/authors/?q=ai:van.au-vo"Nguyen, Huy Tuan"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yongSummary: This paper investigates an inverse problem for fractional Rayleigh-Stokes equations with nonlinear source. The fractional derivative in time is taken in the sense of Riemann-Liouville. The proposed problem has many applications in some non-Newtonian fluids. We obtain some results on the existence and regularity of mild solutions.Boundary determination of electromagnetic and Lamé parameters with corrupted datahttps://zbmath.org/1472.354492021-11-25T18:46:10.358925Z"Caro, Pedro"https://zbmath.org/authors/?q=ai:caro.pedro"Lai, Ru-Yu"https://zbmath.org/authors/?q=ai:lai.ru-yu"Lin, Yi-Hsuan"https://zbmath.org/authors/?q=ai:lin.yi-hsuan"Zhou, Ting"https://zbmath.org/authors/?q=ai:zhou.tingSummary: We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lamé moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.On an inverse problem to a mixed problem for the Poisson equationhttps://zbmath.org/1472.354502021-11-25T18:46:10.358925Z"Chernikova, N. Yu."https://zbmath.org/authors/?q=ai:chernikova.natalia-yu"Laneev, E. B."https://zbmath.org/authors/?q=ai:laneev.eugeniy-b"Muratov, M. N."https://zbmath.org/authors/?q=ai:muratov.m-n"Ponomarenko, E. Yu."https://zbmath.org/authors/?q=ai:ponomarenko.e-yuSummary: The stable solution to the inverse problem of restoring the function of density distribution of the sources, corresponding to an infinitely thin body, was acquired in mixed boundary problem of Poisson equation. The Tikhonov method of regularization using principle of minimum smoothing was applied for obtaining the stable solution.
For the entire collection see [Zbl 1467.34001].A variational inequality based stochastic approximation for inverse problems In stochastic partial differential equationshttps://zbmath.org/1472.354512021-11-25T18:46:10.358925Z"Hawks, Rachel"https://zbmath.org/authors/?q=ai:hawks.rachel"Jadamba, Baasansuren"https://zbmath.org/authors/?q=ai:jadamba.baasansuren"Khan, Akhtar A."https://zbmath.org/authors/?q=ai:khan.akhtar-ali"Sama, Miguel"https://zbmath.org/authors/?q=ai:sama.miguel"Yang, Yidan"https://zbmath.org/authors/?q=ai:yang.yidanSummary: The primary objective of this work is to study the inverse problem of identifying a parameter in partial differential equations with random data. We explore the nonlinear inverse problem in a variational inequality framework. We propose a projected-gradient-type stochastic approximation scheme for general variational inequalities and give a complete convergence analysis under weaker conditions on the random noise than those commonly imposed in the available literature. The proposed iterative scheme is tested on the inverse problem of parameter identification. We provide a derivative characterization of the solution map, which is used in computing the derivative of the objective map. By employing a finite element based discretization scheme, we derive the discrete formulas necessary to test the developed stochastic approximation scheme. Preliminary numerical results show the efficacy of the developed framework.
For the entire collection see [Zbl 1470.49002].Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and sourcehttps://zbmath.org/1472.354522021-11-25T18:46:10.358925Z"Jin, Bangti"https://zbmath.org/authors/?q=ai:jin.bangti"Zhou, Zhi"https://zbmath.org/authors/?q=ai:zhou.zhiOn the identification of the nonlinearity parameter in the Westervelt equation from boundary measurementshttps://zbmath.org/1472.354532021-11-25T18:46:10.358925Z"Kaltenbacher, Barbara"https://zbmath.org/authors/?q=ai:kaltenbacher.barbara"Rundell, William"https://zbmath.org/authors/?q=ai:rundell.williamSummary: We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as \(B/A\) in the literature which is part of a space dependent coefficient \(\kappa\) in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set \(\Sigma\) representing the receiving transducer array. After an analysis of the map from \(\kappa\) to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover \(\kappa\). Numerical simulations will also be shown to illustrate the efficiency of the methods.Corrigendum to: ``Analysis of regularized inversion of data corrupted by white Gaussian noise''https://zbmath.org/1472.354542021-11-25T18:46:10.358925Z"Kekkonen, Hanne"https://zbmath.org/authors/?q=ai:kekkonen.hanne"Lassas, Matti"https://zbmath.org/authors/?q=ai:lassas.matti-j"Siltanen, Samuli"https://zbmath.org/authors/?q=ai:siltanen.samuliCorrigendum to the authors' paper [ibid. 30, No. 4, Article ID 045009, 18 p. (2014; Zbl 1287.35101)].The uniqueness of inverse problems for a fractional equation with a single measurementhttps://zbmath.org/1472.354552021-11-25T18:46:10.358925Z"Kian, Yavar"https://zbmath.org/authors/?q=ai:kian.yavar"Li, Zhiyuan"https://zbmath.org/authors/?q=ai:li.zhiyuan"Liu, Yikan"https://zbmath.org/authors/?q=ai:liu.yikan"Yamamoto, Masahiro"https://zbmath.org/authors/?q=ai:yamamoto.masahiroSummary: This article is concerned with an inverse problem on simultaneously determining some unknown coefficients and/or an order of derivative in a multidimensional time-fractional evolution equation either in a Euclidean domain or on a Riemannian manifold. Based on a special choice of the Dirichlet boundary input, we prove the unique recovery of at most two out of four \(x\)-dependent coefficients (possibly with an extra unknown fractional order) by a single measurement of the partial Neumann boundary output. Especially, both a vector-valued velocity field of a convection term and a density can also be uniquely determined. The key ingredient turns out to be the time-analyticity of the decomposed solution, which enables the construction of Dirichlet-to-Neumann maps in the frequency domain and thus the application of inverse spectral results.Inverse problems for a generalized subdiffusion equation with final overdeterminationhttps://zbmath.org/1472.354562021-11-25T18:46:10.358925Z"Kinash, Nataliia"https://zbmath.org/authors/?q=ai:kinash.nataliia"Janno, Jaan"https://zbmath.org/authors/?q=ai:janno.jaanSummary: We consider two inverse problems for a generalized subdiffusion equation that use the final overdetermination condition. Firstly, we study a problem of reconstruction of a specific space-dependent component in a source term. We prove existence, uniqueness and stability of the solution to this problem. Based on these results, we consider an inverse problem of identification of a space-dependent coefficient of a linear reaction term. We prove the uniqueness and local existence and stability of the solution to this problem.A linear sampling method for inverse acoustic scattering by a locally rough interfacehttps://zbmath.org/1472.354572021-11-25T18:46:10.358925Z"Li, Jianliang"https://zbmath.org/authors/?q=ai:li.jianliang"Yang, Jiaqing"https://zbmath.org/authors/?q=ai:yang.jiaqing"Zhang, Bo"https://zbmath.org/authors/?q=ai:zhang.boSummary: This paper is concerned with the inverse problem of time-harmonic acoustic scattering by an unbounded, locally rough interface which is assumed to be a local perturbation of a plane. The purpose of this paper is to recover the local perturbation of the interface from the near-field measurement given on a straight line segment with a finite distance above the interface and generated by point sources. Precisely, we propose a novel version of the linear sampling method to recover the location and shape of the local perturbation of the interface numerically. Our method is based on a modified near-field operator equation associated with a special rough surface, constructed by reformulating the forward scattering problem into an equivalent integral equation formulation in a bounded domain, leading to a fast imaging algorithm. Numerical experiments are presented to illustrate the effectiveness of the imaging method.Corrigenda to: ``Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation''https://zbmath.org/1472.354582021-11-25T18:46:10.358925Z"Milne, Tristan"https://zbmath.org/authors/?q=ai:milne.tristan"Mansouri, Abdol-Reza"https://zbmath.org/authors/?q=ai:mansouri.abdol-rezaSummary: The proof of Lemma 4.4 in our article [ibid. 371, No. 12, 8781--8810 (2019; Zbl 1419.35251)] contains a flaw. In proving the existence of a minimizer of the map \(\mathbf{A} \mapsto I_\epsilon [\mathbf{A}]\) defined therein, we stated that this map is a convex function of \(\mathbf{A} \). This is incorrect, as \(I_\epsilon\) is a composition of two convex functions, a quadratic form and an absolute value, and since the absolute value function is not monotonic, there is no guarantee that the resulting functional is convex. This short article corrects this flaw by showing that there is a continuous convex functional \(J_\epsilon\) such that \(I_\epsilon [\mathbf{A}] = J_\epsilon [\mathbf{A}^2]\), and then employing weak lower semi-continuity of \(J_\epsilon\) to demonstrate the existence of a minimizer of \(I_\epsilon \).Hölder stability of quantitative photoacoustic tomography based on partial datahttps://zbmath.org/1472.354592021-11-25T18:46:10.358925Z"Triki, Faouzi"https://zbmath.org/authors/?q=ai:triki.faouzi"Xue, Qi"https://zbmath.org/authors/?q=ai:xue.qiAn inverse problem for a time-fractional advection equation associated with a nonlinear reaction termhttps://zbmath.org/1472.354602021-11-25T18:46:10.358925Z"Vo, Hoang-Hung"https://zbmath.org/authors/?q=ai:vo.hoang-hung"Le Minh, Triet"https://zbmath.org/authors/?q=ai:le-minh.triet"Hong, Phong Luu"https://zbmath.org/authors/?q=ai:hong.phong-luu"Van, Canh Vo"https://zbmath.org/authors/?q=ai:van.canh-voSummary: Fractional derivative is an important notion in the study of the contemporary mathematics not only because it is more mathematically general than the classical derivative but also it really has applications to understand many physical phenomena. In particular, fractional derivatives are related to long power-law particle jumps, which can be understood as transient anomalous sub-diffusion model (see \textit{F. Sabzikar} et al. [J. Comput. Phys. 293, 14--28 (2015; Zbl 1349.26017)]; \textit{I. Sokolov} et al. [``Fractional kinetics'', Phys. Today 55, No. 11, 48--54 (2002; \url{doi:10.1063/1.1535007})]; \textit{J. Klafter} and \textit{I. M. Sokolov} [``Anomalous diffusion spreads its wings'', Phys. World 18, No. 8, 29--32 (2005; \url{doi:10.1088/2058-7058/18/8/33})]; \textit{Y. Zhang} et al. [``Linking fluvial bed sediment transport across scales'', Geophys. Res. Lett. 39, No. 20, Article ID L20404 (2012; \url{doi:10.1029/2012GL053476})]). Based on the models given in \textit{H. Scher} and \textit{E. W. Montroll} [``Anomalous transit-time dispersion in amorphous solids'', Phys. Rev. B (3) 12, No. 6, 2455--2477 (1975; \url{doi:10.1103/PhysRevB.12.2455})] and \textit{G. H. Zheng} and \textit{T. Wei} [Appl. Math. Comput. 218, No. 2, 396--405 (2011; Zbl 1228.65187)], we study an inverse problem for the advection equation with a nonlinear reaction term in a two-dimensional semi-infinite domain for which we recover the initial distribution from the observation data provided at the final location \(x = 1\). This problem is severely ill-posed in the sense of Hadamard. Thus, we propose a regularization method to construct an approximate solution for the problem. From that, convergence rate of the regularized solution is obtained under some a priori bound assumptions on the exact solution. Eventually, a numerical experiment is given to show the effectiveness of the proposed regularization methods.Numerical conductivity reconstruction from partial interior current density information in three dimensionshttps://zbmath.org/1472.354612021-11-25T18:46:10.358925Z"Yazdanian, Hassan"https://zbmath.org/authors/?q=ai:yazdanian.hassan"Knudsen, Kim"https://zbmath.org/authors/?q=ai:knudsen.kim-mTwo-phase free boundary problems in convex domainshttps://zbmath.org/1472.354622021-11-25T18:46:10.358925Z"Beck, Thomas"https://zbmath.org/authors/?q=ai:beck.thomas-l"Jerison, David"https://zbmath.org/authors/?q=ai:jerison.david-s"Raynor, Sarah"https://zbmath.org/authors/?q=ai:raynor.sarahSummary: We study the regularity of minimizers of a two-phase free boundary problem. For a class of \(n\)-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt-Caffarelli-Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland-Hayman inequality for geodesically convex subsets of the sphere with Neumann boundary conditions. To apply this inequality, in addition to convexity, we require a Dini condition governing the rate at which the fixed boundary converges to its limit cone at each boundary point.On the existence, uniqueness, and new analytic approximate solution of the modified error function in two-phase Stefan problemshttps://zbmath.org/1472.354632021-11-25T18:46:10.358925Z"Bougoffa, Lazhar"https://zbmath.org/authors/?q=ai:bougoffa.lazhar"Rach, Randolph C."https://zbmath.org/authors/?q=ai:rach.randolph-c"Mennouni, Abdelaziz"https://zbmath.org/authors/?q=ai:mennouni.abdelazizSummary: This paper provides a new proof of the existence and uniqueness of the solution for a nonlinear boundary value problem
\[
\begin{cases}
[(1+\delta y) y^\prime]^\prime &+ \, 2x(1+ \gamma y)y^\prime = 0, 0 < x < \infty, \\
&y(0) = 0, y(\infty) = 1,
\end{cases}
\] which describes the study of two-phase Stefan problems on the semi-infinite line \([0, \infty)\). This result considerably extends the analysis of a recent work. A highly accurate analytic approximate solution of this problem is also provided via the Adomian decomposition method.Regularity of flat free boundaries for a \(p(x)\)-Laplacian problem with right hand sidehttps://zbmath.org/1472.354642021-11-25T18:46:10.358925Z"Ferrari, Fausto"https://zbmath.org/authors/?q=ai:ferrari.fausto"Lederman, Claudia"https://zbmath.org/authors/?q=ai:lederman.claudia-bSummary: We consider viscosity solutions to a one-phase free boundary problem for the \(p(x)\)-Laplacian with non-zero right hand side. We apply the tools developed in [\textit{D. De Silva}, Interfaces Free Bound. 13, No. 2, 223--238 (2011; Zbl 1219.35372)] to prove that flat free boundaries are \(C^{1,\alpha}\). Moreover, we obtain some new results for the operator under consideration that are of independent interest.Measurability of a solution of a free boundary problem describing adsorption phenomenonhttps://zbmath.org/1472.354652021-11-25T18:46:10.358925Z"Kumazaki, K."https://zbmath.org/authors/?q=ai:kumazaki.kotaSummary: In this paper we consider infinite number of one dimensional free boundary problems as a mathematical model describing adsorption phenomena in holes of a porous material. Here we denote by \(P(x,u_0(x),h(x))\) the free boundary problem for \(x\in\Omega\), where \(x\) is a parameter taking a value in \(\Omega\) and \(u_0(x)\) and \(h(x)\) are the initial data and the boundary data.
In [the author, ibid. 25, 289--305 (2016; Zbl 1472.35466)] the problem was studied and we obtain the continuous property of the solution with respect to \(x\), when \(u_0\) and \(h\) are continuous. The main purpose of this paper is to establish the measurability of the solution with respect to \(x\) under relaxed assumptions given in [loc. cit.] for \(u_0\) and \(h\).Continuous dependence of a solution of a free boundary problem describing adsorption phenomenon for a given datahttps://zbmath.org/1472.354662021-11-25T18:46:10.358925Z"Kumazaki, Kota"https://zbmath.org/authors/?q=ai:kumazaki.kotaSummary: In this paper we consider a one dimensional free boundary problem as a mathematical model describing adsorption phenomenon in one hole of a porous media. This model is proposed by \textit{T. Aiki} et al. [``A mathematical model for a hysteresis appearing in adsorption phenomena'', RIMS Kôkyûroku 1856, 1--11 (2013), \url{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/1856.html}]; \textit{N. Sato} et al. [Netw. Heterog. Media 9, No. 4, 655--668 (2014; Zbl 1310.35246)] and consists of a partial differential equation for the relative humidity in the hole and an ordinary differential equation of the front of water region which represents the growth rate for water region. For this model, Sato [loc. cit] proved the existence and uniqueness of a time local solution, and \textit{T. Aiki} and \textit{Y. Murase} [J. Math. Anal. Appl. 445, No. 1, 837--854 (2017; Zbl 1386.35484)] showed the existence of a solution globally in time and the convergence to a solution of a steady state problem as a large time behavior of solutions. In this paper, we consider this mathematical model in each hole for each position \(x\) of the porous media with respect to \(\Omega\subset\mathbb{R}^3\), and prove the continuous dependance of the solution of this problem with respect to \(x\in\Omega\).Geodesic Loewner paths with varying boundary conditionshttps://zbmath.org/1472.354672021-11-25T18:46:10.358925Z"Mcdonald, Robb"https://zbmath.org/authors/?q=ai:mcdonald.n-robbSummary: Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner's equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped `mathematical' plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of \(\pi /5\).Wong-Zakai approximations of the non-autonomous stochastic Fitzhugh-Nagumo system on \(\mathbb{R}^N\) in higher regular spaceshttps://zbmath.org/1472.354682021-11-25T18:46:10.358925Z"Zhao, Wenqiang"https://zbmath.org/authors/?q=ai:zhao.wenqiangSummary: In this paper, we consider the Wong-Zakai approximations of a non-autonomous stochastic FitzHugh-Nagumo system driven by a multiplicative white noise with an arbitrary intensity. The convergence of solutions of the path-wise deterministic system to that of the corresponding stochastic system is established in higher regular spaces by means of a new iteration technique and an optimal multiplier at different stages. Furthermore, we prove that the random attractor of the path-wise deterministic system converges to that of the non-autonomous stochastic FitzHugh-Nagumo system in higher regular spaces when the size of approximation vanishes, with much looser conditions on the nonlinearity.
{\copyright 2021 American Institute of Physics}Infinite order \(\Psi\mathrm{DOs}\): composition with entire functions, new Shubin-Sobolev spaces, and index theoremhttps://zbmath.org/1472.354692021-11-25T18:46:10.358925Z"Pilipović, Stevan"https://zbmath.org/authors/?q=ai:pilipovic.stevan-r"Prangoski, Bojan"https://zbmath.org/authors/?q=ai:prangoski.bojan"Vindas, Jasson"https://zbmath.org/authors/?q=ai:vindas.jassonSummary: We study global regularity and spectral properties of power series of the Weyl quantisation \(a^w\), where \(a(x,\xi)\) is a classical elliptic Shubin polynomial. For a suitable entire function \(P\), we associate two natural infinite order operators to \(a^w\), \(P(a^w)\) and \((P\circ a)^w\), and prove that these operators and their lower order perturbations are globally Gelfand-Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to \(\infty\) for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of \(f\)-\(\Gamma^{*,\infty}_{A_p,\rho}\)-elliptic symbols, where \(f\) is a function of ultrapolynomial growth and \(\Gamma^{*,\infty}_{A_p,\rho}\) is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov-Hörmander integral formula.Configuration sets with nonempty interiorhttps://zbmath.org/1472.354702021-11-25T18:46:10.358925Z"Greenleaf, Allan"https://zbmath.org/authors/?q=ai:greenleaf.allan"Iosevich, Alex"https://zbmath.org/authors/?q=ai:iosevich.alex"Taylor, Krystal"https://zbmath.org/authors/?q=ai:taylor.krystalSummary: A theorem of Steinhaus states that if \(E\subset\mathbb{R}^d\) has positive Lebesgue measure, then the difference set \(E-E\) contains a neighborhood of 0. Similarly, if \(E\) merely has Hausdorff dimension \(\dim_{\mathcal{H}}(E)>(d+1)/2\), a result of Mattila and Sjölin states that the distance set \(\varDelta(E)\subset\mathbb{R}\) contains an open interval. In this work, we study such results from a general viewpoint, replacing \(E-E\) or \(\varDelta (E)\) with more general \(\varPhi\)-configurations for a class of \(\varPhi:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}^k\), and showing that, under suitable lower bounds on \(\dim_{\mathcal{H}}(E)\) and a regularity assumption on the family of generalized Radon transforms associated with \(\varPhi\), it follows that the set \(\varDelta_\varPhi(E)\) of \(\varPhi\)-configurations in \(E\) has nonempty interior in \(\mathbb{R}^k\). Further extensions hold for \(\varPhi\)-configurations generated by two sets, \(E\) and \(F\), in spaces of possibly different dimensions and with suitable lower bounds on \(\dim_{\mathcal{H}}(E)+\dim_{\mathcal{H}}(F)\).Continuum limits of pluri-Lagrangian systemshttps://zbmath.org/1472.370682021-11-25T18:46:10.358925Z"Vermeeren, Mats"https://zbmath.org/authors/?q=ai:vermeeren.matsSummary: A pluri-Lagrangian (or Lagrangian multiform) structure is an attribute of integrability that has mainly been studied in the context of multidimensionally consistent lattice equations. It unifies multidimensional consistency with the variational character of the equations. An analogous continuous structure exists for integrable hierarchies of differential equations. We present a continuum limit procedure for pluri-Lagrangian systems. In this procedure, the lattice parameters are interpreted as Miwa variables, describing a particular embedding in continuous multi-time of the mesh on which the discrete system lives. Then, we seek differential equations whose solutions interpolate the embedded discrete solutions. The continuous systems found this way are hierarchies of differential equations. We show that this continuum limit can also be applied to the corresponding pluri-Lagrangian structures. We apply our method to the discrete Toda lattice and to equations H1 and Q1$_{\delta=0}$ from the ABS list.Modified constrained KP hierarchy and bi-Hamiltonian structureshttps://zbmath.org/1472.370702021-11-25T18:46:10.358925Z"Li, Hongmin"https://zbmath.org/authors/?q=ai:li.hongminThe author presents a new class of KP-like integrable hierarchy, called {modified constrained KP hierarchy}. It differs from the standard constrained KP hierarchy in [\textit{W. Oevel} and \textit{W. Strampp}, Commun. Math. Phys. 157, No. 1, 51--81 (1993; Zbl 0793.35095)] by the addition of a spectral parameter. The constraint is given by \[ \hat{L}=\partial^n-u_{n-2}\partial^{n-2}-\cdots-u_1\partial-u_0-\lambda\,q\partial^{-1}r, \] where \(q\) and \(r\) are, the eigenfunctions of the scattering problem for a KP hierarchy. The standard constrained KP hierarchy is recovered with \(\lambda=1\).
The author presents the hierarchies obtained in the case \(n=1,2,3\) and their bi-Hamiltonian structures. Despite the identification of a gauge transform linking the \(n=2\) and \(n=3\) cases, the Yajima-Oikawa and Melnikov hierarchies (and their bi-Hamiltonian structures as well) are not obtained by Miura transforms, which would have been very cumbersome, but using a trace identity.Lie symmetry analysis and conservation laws of a two-wave mode equation for the integrable Kadomtsev-Petviashvili equationhttps://zbmath.org/1472.370712021-11-25T18:46:10.358925Z"Moretlo, T. S."https://zbmath.org/authors/?q=ai:moretlo.t-s"Muatjetjeja, B."https://zbmath.org/authors/?q=ai:muatjetjeja.ben"Adem, A. R."https://zbmath.org/authors/?q=ai:adem.abdullahi-rashidSummary: Lie symmetry analysis is performed on a two-wave mode equation for the integrable Kadomtsev-Petviashvili (TKP) equation which describes the propagation of two different wave modes in the same direction simultaneously. The similarity reductions and an exact solution are computed. In addition to this, we derive the conservation laws for the underlying equation.The Fokas method for integrable evolution equations on a time-dependent intervalhttps://zbmath.org/1472.370752021-11-25T18:46:10.358925Z"Xia, Baoqiang"https://zbmath.org/authors/?q=ai:xia.baoqiangSummary: We demonstrate how to use the Fokas method to analyze initial-boundary value problems for integrable evolution equations posed on a time-dependent interval. In particular, we implement this method to a general dispersive linear evolution equation with spatial derivatives of arbitrary order and to the nonlinear Schrödinger equation, formulated in the time-dependent domain \(l_{1}(t)<x<l_{2}(t),\; 0<t<T,\; \text{where}\; l_{1}(t)\; \text{and}\; l_{2}(t)\) are given, real, twice differentiable functions whose first derivatives are monotonic, and \(T\) is a positive fixed constant.{
\copyright 2019 American Institute of Physics}Chaotic-like transfers of energy in Hamiltonian PDEshttps://zbmath.org/1472.370792021-11-25T18:46:10.358925Z"Giuliani, Filippo"https://zbmath.org/authors/?q=ai:giuliani.filippo"Guardia, Marcel"https://zbmath.org/authors/?q=ai:guardia.marcel"Martin, Pau"https://zbmath.org/authors/?q=ai:martin.pau"Pasquali, Stefano"https://zbmath.org/authors/?q=ai:pasquali.stefanoThe authors study the following problems: \begin{eqnarray*} u_{tt} - \Delta u + u^3 & = & 0, \quad u=u(t,x), \quad t \in \mathbb{R}, \quad x \in \mathbb{T}^2 \\
u_{tt} + \Delta^2 u + u^3 & = & 0, \quad u=u(t,x), \quad t \in \mathbb{R}, \quad x \in \mathbb{T}^2. \end{eqnarray*} They prove the existence of special beating solutions, namely solutions that exhibit transfer of energy between Fourier modes.Global stability of traveling waves for a spatially discrete diffusion system with time delayhttps://zbmath.org/1472.390102021-11-25T18:46:10.358925Z"Liu, Ting"https://zbmath.org/authors/?q=ai:liu.ting"Zhang, Guo-Bao"https://zbmath.org/authors/?q=ai:zhang.guobao.1|zhang.guobaoSummary: This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in \( L^\infty(\mathbb{R})\times L^\infty(\mathbb{R}) \) with the exponential convergence rate \( e^{-\mu t}\) for some constant \( \mu>0 \).A series representation of the discrete fractional Laplace operator of arbitrary orderhttps://zbmath.org/1472.390382021-11-25T18:46:10.358925Z"Frugé Jones, Tiffany"https://zbmath.org/authors/?q=ai:fruge-jones.tiffany"Kostadinova, Evdokiya Georgieva"https://zbmath.org/authors/?q=ai:kostadinova.evdokiya-georgieva"Padgett, Joshua Lee"https://zbmath.org/authors/?q=ai:padgett.joshua-lee"Sheng, Qin"https://zbmath.org/authors/?q=ai:sheng.qinSummary: Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.A comparison principle for convolution measures with applicationshttps://zbmath.org/1472.420122021-11-25T18:46:10.358925Z"Oliveira e Silva, Diogo"https://zbmath.org/authors/?q=ai:oliveira-e-silva.diogo"Quilodrán, René"https://zbmath.org/authors/?q=ai:quilodran.reneSummary: We establish the general form of a geometric comparison principle for \(n\)-fold convolutions of certain singular measures in \(\mathbb{R}^d\) which holds for arbitrary \(n\) and \(d\). This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [\textit{G. Brocchi} et al., Anal. PDE 13, No. 2, 477--526 (2020; Zbl 1435.35015)].Multiplier theorems via martingale transformshttps://zbmath.org/1472.420132021-11-25T18:46:10.358925Z"Bañuelos, Rodrigo"https://zbmath.org/authors/?q=ai:banuelos.rodrigo"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabrice"Chen, Li"https://zbmath.org/authors/?q=ai:chen.li.2|chen.li.4|chen.li.5|chen.li.6|chen.li.7|chen.li.1|chen.li.3"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannickSummary: We develop a new and general approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by \textit{P. R. Stinga} and \textit{J. L. Torrea} [Commun. Partial Differ. Equations 35, No. 10--12, 2092--2122 (2010; Zbl 1209.26013)], which is of independent interest. Our methods here also recover the sharp \(L^p\) bounds for second order Riesz transforms by a limiting argument.Norm comparison estimates for the composite operatorhttps://zbmath.org/1472.420322021-11-25T18:46:10.358925Z"Li, Xuexin"https://zbmath.org/authors/?q=ai:li.xuexin"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.10|wang.yong.9|wang.yong.8|wang.yong.7"Xing, Yuming"https://zbmath.org/authors/?q=ai:xing.yumingSummary: This paper obtains the Lipschitz and BMO norm estimates for the composite operator \(\mathbb M_s \circ P\) applied to differential forms. Here, \(\mathbb M_3\) is the Hardy-Littlewood maximal operator, and \(P\) is the potential operator. As applications, we obtain the norm estimates for the Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.On the Helmholtz decompositions of vector fields of bounded mean oscillation and in real Hardy spaces over the half spacehttps://zbmath.org/1472.420362021-11-25T18:46:10.358925Z"Giga, Yoshikazu"https://zbmath.org/authors/?q=ai:giga.yoshikazu"Gu, Zhongyang"https://zbmath.org/authors/?q=ai:gu.zhongyangSummary: This paper is concerned with the Helmholtz decompositions of vector fields of bounded mean oscillation over the half space and vector fields in real Hardy spaces over the half space. It proves the Helmholtz decomposition for vector fields of bounded mean oscillation over the half space whereas a partial Helmholtz decomposition for vector fields in real Hardy spaces over the half space. Meanwhile, it also establishes two sets of theories of real Hardy spaces over the half space which are compatible with the theory of \textit{A. Miyachi} [Stud. Math. 96, No. 3, 205--228 (1990; Zbl 0716.42017)].Lebedev-Skalskaya transforms on certain function spaces and associated pseudo-differential operatorshttps://zbmath.org/1472.440022021-11-25T18:46:10.358925Z"Mandal, U. K."https://zbmath.org/authors/?q=ai:mandal.upain-kumar"Prasad, Akhilesh"https://zbmath.org/authors/?q=ai:prasad.akhileshThe paper presents one of the index transforms known as the Lebedev-Skalskaya transform (LS-transform) given by \textit{S.~B. Yakubovich} in [Index transforms. London: World Scientific (1996; Zbl 0845.44001)] whose kernel is the modified Bessel function of the second kind. The authors of the present paper reflect the recent developments for the LS transform given by many researchers which can be seen in the exhaustive list of references.
This paper consists of five sections. Section~1 presents the introduction of the LS transform and its inversion. Two differential operators (see (1.22) and (1.23)) are introduced. The basic properties such as Plancherel's and Parseval's relation corresponding to the LS transform and adjoint LS transform are shown. Employing the translation operator, the convolution for the LS transform and other related properties are discussed. The operational formulas associated with the differential operator, translation and mainly convolution operator are represented in Section~2, and they are estimated in Lebesgue spaces for the LS transform. The continuity of the LS transform in the Lebesgue space is shown and then related to some other function spaces, dealt with in Section~3. The pseudodifferential operators in terms of the LS transform are defined in Section~4 and studied further for the integral representation of pseudodifferential operators.
Section~5 is devoted to an application. Using the LS transform, an integral equation with convolution kernel and an initial value problem are solved.\(L^q\)-solvability for an equation of viscoelasticity in power type materialhttps://zbmath.org/1472.450022021-11-25T18:46:10.358925Z"de Andrade, Bruno"https://zbmath.org/authors/?q=ai:de-andrade.bruno"Silva, Clessius"https://zbmath.org/authors/?q=ai:silva.clessius"Viana, Arlúcio"https://zbmath.org/authors/?q=ai:viana.arlucioThe authors study the existence, uniqueness, regularity, continuous dependence, unique continuation, a blow-up alternative for mild solutions, and global well-posedness of the nonlinear Volterra equation \[ u_t = \int_0^t dg_\alpha(s) \Delta u(t-s,x)- \nabla p +h - (u\cdot \nabla u),\quad \textrm{div}(u)=0, \] in \((0,\infty)\times \Omega\), where \(u=0\) on \((0,\infty)\times \partial \Omega\) and \(u(0,x)=u_0(x)\) in \(\Omega\). Here the kernel is taken to be \(g_\alpha (t)= t^{\alpha}/\Gamma(\alpha+1)\) with \(0\leq \alpha <1\) and a mild solution is a solution to the equation \[ u(t)= S_\alpha (tA)u_0 + \int_0^t S_\alpha((t-s)A)(F(u)(s)+Ph(s))\, ds, \] where \(P\) is the Leray projection on divergence free functions, \(F(u)= P(u\cdot \nabla)u\), \(A=P\Delta\) and \[S_\alpha(tA)= \frac 1{2\pi i} \int_{Ha}e^{\lambda t}\lambda^\alpha(\lambda^{\alpha+1}I+A)^{-1}\, d\lambda,\] where \(t>0\) and \(Ha\) is a suitable path.
The existence results show that the mild solutions have more spatial regularity in terms of estimates on norms in fractional power spaces when \(\alpha\) is closer to \(0\), the case of the Navier-Stokes equations. The linear estimates needed are stated in an abstract setting for sectorial operators which makes it possible to restate the results for some other equations as well.Time memory effect in entropy decay of Ornstein-Uhlenbeck operatorshttps://zbmath.org/1472.450092021-11-25T18:46:10.358925Z"Agresti, Antonio"https://zbmath.org/authors/?q=ai:agresti.antonio"Loreti, Paola"https://zbmath.org/authors/?q=ai:loreti.paola"Sforza, Daniela"https://zbmath.org/authors/?q=ai:sforza.danielaSummary: We investigate the effect of memory terms on the entropy decay of the solutions to diffusion equations with Ornstein-Uhlenbeck operators. Our assumptions on the memory kernels include Caputo-Fabrizio operators and, more generally, the stretched exponential functions. We establish a sharp rate decay for the entropy. Examples and numerical simulations are also given to illustrate the results.Shape holomorphy of the Calderón projector for the Laplacian in \(\mathbb{R}^2\)https://zbmath.org/1472.450112021-11-25T18:46:10.358925Z"Henríquez, Fernando"https://zbmath.org/authors/?q=ai:henriquez.fernando"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christophThe authors establish the holomorphic dependence of the Calderón projector for the Laplace equation on a collection of sufficiently smooth Jordan curves in the Cartesian Euclidean plan. To be precise, they establish holomorphy of the domain-to-operator map associated to the Calderón projector.A Pólya-Szegő principle for general fractional Orlicz-Sobolev spaceshttps://zbmath.org/1472.460282021-11-25T18:46:10.358925Z"De Nápoli, Pablo"https://zbmath.org/authors/?q=ai:de-napoli.pablo-luis"Fernández Bonder, Julián"https://zbmath.org/authors/?q=ai:fernandez-bonder.julian"Salort, Ariel"https://zbmath.org/authors/?q=ai:salort.ariel-martinSummary: In this article, we prove modular and norm Pólya-Szegő inequalities in general fractional Orlicz-Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of these spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary, we prove a Rayleigh-Faber-Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.Fractional Orlicz-Sobolev embeddingshttps://zbmath.org/1472.460342021-11-25T18:46:10.358925Z"Alberico, Angela"https://zbmath.org/authors/?q=ai:alberico.angela"Cianchi, Andrea"https://zbmath.org/authors/?q=ai:cianchi.andrea"Pick, Luboš"https://zbmath.org/authors/?q=ai:pick.lubos"Slavíková, Lenka"https://zbmath.org/authors/?q=ai:slavikova.lenkaSummary: The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in \(\mathbb{R}^n\). An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all rearrangement-invariant spaces, is also established. Both spaces of order \(s\in(0, 1)\), and higher-order spaces are considered. Related Hardy type inequalities are proposed as well.Traces on operator ideals and related linear forms on sequence ideals. IVhttps://zbmath.org/1472.470162021-11-25T18:46:10.358925Z"Pietsch, Albrecht"https://zbmath.org/authors/?q=ai:pietsch.albrechtThe author continues in this paper a sequence of papers on the topic. Here, he deals with dyadic representations of a bounded linear operator \(S\in \mathcal L(X,Y)\), meaning that \(S=\sum_{k=0}^\infty S_k\) where \(S_k\in \mathcal L(X,Y)\) are finite rank operators with \(\operatorname{rank} (S_k)\leq 2^k\) and with shift-monotone sequences ideals \(\mathcal \xi(\mathbb N_0)\), that is, \(S_+\)-invariant linear subspaces of \(\ell_\infty(\mathbb N_0)\), where \(S_+\) stands for the forward shift, satisfying that, if \(a\in \mathcal \xi(\mathbb N_0)\), \(b\in \ell_\infty(\mathbb N_0)\) and \(\sup_{n\ge k}|b_n|\le \sup_{n\ge k}|a_n|\), then \(a\in \mathcal \xi(\mathbb N_0)\). The author refers to \(S=\sum_{k=0}^\infty S_k\) as an \((\mathcal U, \mathcal \xi)\)-representation if \((\|S-\sum_{k=0}^{n-1} S_k\|_{\mathcal U})_n\in \mathbb \xi(\mathbb N_0)\).
It is known [the author, Indag. Math., New Ser. 25, No. 2, 341--365 (2014; Zbl 1319.47067)] that there is a one-to-one correspondence between all symmetric sequence ideals and all shift-monotone sequence ideals. The correspondence maps to each shift-monotone sequence ideal \(\mathcal \xi (\mathbb N_0)\) the symmetric sequence ideal \(s(\mathbb N_0)=\{a\in \ell_\infty(\mathbb N_0): (a_{2^k})\in \mathcal \xi(\mathbb N_0)\}\) and the sequence ideals \(s(\mathbb N_0)\) and \(\xi(\mathbb N_0)\) are said to be associated.
The main result establishes that, if \(\mathcal U\) is a quasi Banach operator ideal of trace type \(\rho\), that is, \(|\operatorname{trace}(F)|\le c n^\rho\|F\|_{\mathcal U}\) whenever \(F\) is a finite rank operator with \(\operatorname{rank}(F)\le n\) and \(c\) is a constant independent of \(F\), and \(\mu\) is a \(2^{-\rho}S_{+}\)-invariant linear form on a shift-monotone sequence ideal \(\xi(\mathbb N_0)\), then \(\tau(S):=\mu(2^{-k\rho}\operatorname{trace}(S_k))\) does not depend on the choice of the \((\mathcal U, \xi)\)-representation \(S=\sum_{k=0}^\infty S_k\). This result is then applied to concrete examples of Banach operator ideals, such as those given by absolutely \((q,2)\)-summing operators and shift-monotone sequence ideals such as those associated to Lorentz sequence ideals. Concrete examples of operators such as convolution operators generated by functions in certain Lipschitz and Besov classes are provided.
For Parts I--III, see [the author, Indag. Math., New Ser. 25, No. 2, 341--365 (2014; Zbl 1319.47067); Integral Equations Oper. Theory 79, No. 2, 255--299 (2014; Zbl 1337.47031); J. Math. Anal. Appl. 421, No. 2, 971--981 (2015; Zbl 1328.47021)].
For the entire collection see [Zbl 1367.47005].Spectral enclosures for non-self-adjoint extensions of symmetric operatorshttps://zbmath.org/1472.470182021-11-25T18:46:10.358925Z"Behrndt, Jussi"https://zbmath.org/authors/?q=ai:behrndt.jussi"Langer, Matthias"https://zbmath.org/authors/?q=ai:langer.matthias"Lotoreichik, Vladimir"https://zbmath.org/authors/?q=ai:lotoreichik.vladimir"Rohleder, Jonathan"https://zbmath.org/authors/?q=ai:rohleder.jonathanIn the description of many quantum mechanical systems, operators appear as a consequence of heuristic arguments which suggest in a first step a formal expression for the Hamiltonian or Schrödinger operator describing the model. These operators \(S\) are typically unbounded and symmetric on a domain \(\operatorname{dom}{S}\) which is a dense subspace of a Hilbert space \(\mathcal{H}\). In a second crucial step for the description of the quantum mechanical system, one has to choose a closed (in many cases self-adjoint) extension of \(S\) in order to start the analysis of the model. Typically, fixing an extension means to specify the relevant boundary conditions for the system. The paper under review focuses on the description of closed non-selfadjoint extensions of \(S\) which appear as restrictions of the adjoint operator \(S^*\) and on the analysis of some of their spectral properties. The article also presents in its final part several applications of their results to elliptic operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with \(\delta\)-interactions, and to quantum graphs with non-self-adjoint vertex couplings.
In the first part of the article, the authors use an abstract and systematic approach to the description of extensions in terms of so-called boundary triples \((\Gamma_0,\Gamma_1,\mathcal{G})\), where \(\Gamma_{0,1}: \operatorname{dom}S^*\to \mathcal{G}\) satisfy a Green identity on the auxiliary Hilbert space \(\mathcal{G}\). Boundary triples (or their generalizations called quasi boundary triples) provide a useful technique to describe extensions encoding abstractly the boundary data of the problem. To formulate their main results in this part (e.g., Theorem 3.1), the authors use the Weyl function, which is an operator-valued function on the auxiliary Hilbert space defined in terms of the boundary triples. In addition, they introduce also a boundary operator \(B\) (in general non-symmetric) which serves to label the different extensions.
The article has an informative and well-written introduction to the topic describing their methods and results, but also introducing the reader to the literature and alternative approaches in this very active field. The bibliography list contains more than 130 references.Wave front sets with respect to the iterates of an operator with constant coefficientshttps://zbmath.org/1472.470352021-11-25T18:46:10.358925Z"Boiti, C."https://zbmath.org/authors/?q=ai:boiti.chiara"Jornet, D."https://zbmath.org/authors/?q=ai:jornet.david"Juan-Huguet, J."https://zbmath.org/authors/?q=ai:juan-huguet.jordiSummary: We introduce the wave front set \(\text{WF}_*^P(u)\) with respect to the iterates of a hypoelliptic linear partial differential operator with constant coefficients of a classical distribution \(u \in \mathcal{D}'(\Omega)\) in an open set \(\Omega\) in the setting of ultradifferentiable classes of \textit{R. W. Braun} et al. [Result. Math. 17, No. 3--4, 206--237 (1990; Zbl 0735.46022)]. We state a version of the microlocal regularity theorem of Hörmander for this new type of wave front set and give some examples and applications of the former result.Wegner estimate for random divergence-type operators monotone in the randomnesshttps://zbmath.org/1472.470362021-11-25T18:46:10.358925Z"Dicke, Alexander"https://zbmath.org/authors/?q=ai:dicke.alexanderSummary: In this note, a Wegner estimate for random divergence-type operators that are monotone in the randomness is proven. The proof is based on a recently shown unique continuation estimate for the gradient and the ensuing eigenvalue liftings. The random model which is studied here contains quite general random perturbations, among others, some that have a non-linear dependence on the random parameters.An invitation to optimal transport, Wasserstein distances, and gradient flowshttps://zbmath.org/1472.490012021-11-25T18:46:10.358925Z"Figalli, Alessio"https://zbmath.org/authors/?q=ai:figalli.alessio"Glaudo, Federico"https://zbmath.org/authors/?q=ai:glaudo.federicoThis graduate text offers a relatively self-contained introduction to the optimal transport theory. It consists of five chapters and two appendices.
Chapter 1 gives a brief review of the optimal transport theory, recalls certain of basics of measure theory and Riemannian geometry, and shows three typical examples of the transport maps in connection to the classical isoperimetry.
Chapter 2 presents the so-called core of the optimal transport theory: the solution to Kantorovich's problem for general costs; the duality theory; the solution to Monge's problem for suitable costs.
Chapter 3 utilizes the \([1,\infty)\ni p\)-Wasserstein distances to handle an essential relationship among the optimal transport theory, gradient flows in the Hilbert spaces, and partial differential equations.
Chapter 4 shows a differential viewpoint of the optimal transport theory via studying Benamou-Brenier's and Otto's formulas based on the probability measures.
Chapter 5 suggests several applied topics of the optimal transport theory.
Appendix A includes a set of some interesting exercises and their solutions.
Appendix B outlines a proof of the disintegration theorem.Boundary control for optimal mixing via Navier-Stokes flowshttps://zbmath.org/1472.490052021-11-25T18:46:10.358925Z"Hu, Weiwei"https://zbmath.org/authors/?q=ai:hu.weiwei"Wu, Jiahong"https://zbmath.org/authors/?q=ai:wu.jiahongOptimal control for the infinity obstacle problemhttps://zbmath.org/1472.490072021-11-25T18:46:10.358925Z"Mawi, Henok"https://zbmath.org/authors/?q=ai:mawi.henok"Ndiaye, Cheikh Birahim"https://zbmath.org/authors/?q=ai:ndiaye.cheikh-birahimThe authors investigate an \(\infty\)-obstacle \(f\) optimal control problem \[ W_g^{1,2} (\Omega) = \{ u \in W^{1,2}(\Omega): \; u=g \; \text {on} \; \partial \Omega \} \] consists of minimizing the Dirichlet energy \( \int_\Omega |Du(x)|^2 dx\) over the set
\[ \mathbb{K}_{f,g}^2 = \{ u \in W^{1,2}(\Omega): \; u(x) \ge f(x) \; \text{in} \; \Omega \} \] where \(\Omega \in \mathbb{R}^n\) is a bounded and smooth domain, \(Du\) is the gradient of \(u\), and \(g \in \text{tr}(W^{1,2}(\Omega))\) with tr the trace operator. This optimal control problem models the equilibrium position of an elastic membrane whose boundary is held fixed at \(g\) and is forced to remain above the given obstacle \(f\). It was shown that the optimal control is also an optimal state. Moreover, it was proved that the minimal value of an optimal control problem for the finite obstacle converges to the minimal value of the optimal control problem for the infinite obstacle.Singular optimal control problems for doubly nonlinear and quasi-variational evolution equationshttps://zbmath.org/1472.490092021-11-25T18:46:10.358925Z"Kenmochi, Nobuyuki"https://zbmath.org/authors/?q=ai:kenmochi.nobuyuki"Shirakawa, Ken"https://zbmath.org/authors/?q=ai:shirakawa.ken"Yamazaki, Noriaki"https://zbmath.org/authors/?q=ai:yamazaki.noriakiSummary: Doubly nonlinear and quasi-variational evolution equations governed by double time-dependent subdifferentials are treated in uniformly convex Banach spaces. We establish some abstract results on the existence-uniqueness of solutions together with related optimal control problems in cases when, in general, the state equations have multiple solutions. In this paper, we propose a general class of singular optimal control problems that are set up for non-well-posed state systems. Moreover, we establish an approximation procedure for such singular optimal control problems and discuss some applications.Solvability and optimal controls of semilinear Riemann-Liouville fractional differential equationshttps://zbmath.org/1472.490132021-11-25T18:46:10.358925Z"Pan, Xue"https://zbmath.org/authors/?q=ai:pan.xue"Li, Xiuwen"https://zbmath.org/authors/?q=ai:li.xiuwen"Zhao, Jing"https://zbmath.org/authors/?q=ai:zhao.jing.1|zhao.jing.2|zhao.jing|zhao.jing.3Summary: We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.Correction to: ``On nonlocal variational and quasi-variational inequalities with fractional gradient''https://zbmath.org/1472.490212021-11-25T18:46:10.358925Z"Rodrigues, José Francisco"https://zbmath.org/authors/?q=ai:rodrigues.jose-francisco"Santos, Lisa"https://zbmath.org/authors/?q=ai:santos.lisaCorrection to the authors' paper [ibid. 80, No. 3, 835--852 (2019; Zbl 1429.49011)].On the supremal version of the Alt-Caffarelli minimization problemhttps://zbmath.org/1472.490452021-11-25T18:46:10.358925Z"Crasta, Graziano"https://zbmath.org/authors/?q=ai:crasta.graziano"Fragalà, Ilaria"https://zbmath.org/authors/?q=ai:fragala.ilariaThe authors consider the free boundary problem (P)\(_\Lambda\): \[m_\Lambda:=\min\{J_\Lambda(u):=\|\nabla u\|_\infty+\Lambda|\{u>0\}|:u\in \text{Lip}_1(\Omega)\},\] where \(\Lambda>0,\ |\{u>0\}|\) is the Lebesgue measure of the set \(\{x\in \Omega:u(x)>0\}\) and \[\text{Lip}_1(\Omega):= \{u\in W^{1,\infty}(\Omega)\,:u\ge 0\ \text{in}\ \Omega, \ u=1\ \text{on}\ \partial\Omega\}\] The main results include the existence and uniqueness of the non-constant solution on convex domains, the identification and the geometrical characterization of the variational infinity Bernoulli constant \[\Lambda_{\Omega,\infty}:=\inf\{\Lambda>0:\,(P)_\Lambda\ \text{admits a non-constant solution}\}.\]Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEshttps://zbmath.org/1472.490462021-11-25T18:46:10.358925Z"Guan, Hongbo"https://zbmath.org/authors/?q=ai:guan.hongbo"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong|wang.yong.7|wang.yong.9|wang.yong.5|wang.yong.6|wang.yong.3|wang.yong.8|wang.yong.10|wang.yong.1|wang.yong.2"Zhu, Huiqing"https://zbmath.org/authors/?q=ai:zhu.huiqingSummary: In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.Periodic solutions to the optimal control problem of rotation of a rigid body using internal masshttps://zbmath.org/1472.490492021-11-25T18:46:10.358925Z"Shmatkov, A. M."https://zbmath.org/authors/?q=ai:shmatkov.a-mSummary: A two-dimensional problem of time-optimal rotation of a mechanical system consisting of a rigid body and a mass point is considered. The mass point interacts with the body by internal forces only. The periodic optimal trajectories of the mass point passing through the rigid body center of inertia are found.Viscosity solutions for controlled McKean-Vlasov jump-diffusionshttps://zbmath.org/1472.490542021-11-25T18:46:10.358925Z"Burzoni, Matteo"https://zbmath.org/authors/?q=ai:burzoni.matteo"Ignazio, Vincenzo"https://zbmath.org/authors/?q=ai:ignazio.vincenzo"Reppen, A. Max"https://zbmath.org/authors/?q=ai:reppen.a-max"Soner, H. M."https://zbmath.org/authors/?q=ai:soner.halil-meteThe paper deals with a class of nonlinear integro-differential equations on a subspace of all probability measures on the real line related to the optimal control of McKean-Vlasov jump-diffusions.
The authors investigated an intrinsic notion of viscosity solutions that does not rely on the lifting to a Hilbert space and proved a comparison theorem for these solutions.Asymptotic analysis and topological derivative for 3D quasi-linear magnetostaticshttps://zbmath.org/1472.490612021-11-25T18:46:10.358925Z"Gangl, Peter"https://zbmath.org/authors/?q=ai:gangl.peter"Sturm, Kevin"https://zbmath.org/authors/?q=ai:sturm.kevinSummary: In this paper we study the asymptotic behaviour of the quasilinear curl-curl equation of 3D magnetostatics with respect to a singular perturbation of the differential operator and prove the existence of the topological derivative using a Lagrangian approach. We follow the strategy proposed in [\textit{P. Gangl} and \textit{K. Sturm}, ESAIM, Control Optim. Calc. Var. 26, Paper No. 106, 20 p. (2020; Zbl 1459.49027)] where a systematic and concise way for the derivation of topological derivatives for quasi-linear elliptic problems in \(H^1\) is introduced. In order to prove the asymptotics for the state equation we make use of an appropriate Helmholtz decomposition. The evaluation of the topological derivative at any spatial point requires the solution of a nonlinear transmission problem. We discuss an efficient way for the numerical evaluation of the topological derivative in the whole design domain using precomputation in an offline stage. This allows us to use the topological derivative for the design optimization of an electrical machine.The area minimizing problem in conformal cones. IIhttps://zbmath.org/1472.490652021-11-25T18:46:10.358925Z"Gao, Qiang"https://zbmath.org/authors/?q=ai:gao.qiang"Zhou, Hengyu"https://zbmath.org/authors/?q=ai:zhou.hengyuIn this paper, the authors continue to study the area minimizing problem with prescribed boundary in a class of conformal cones similar to the one published by the authors in [J. Funct. Anal. 280, No. 3, Article ID 108827, 40 p. (2021; Zbl 1461.49056)]. If \(N\) is an \(n\)-dimensional open Riemannian manifold with a metric \(\sigma\), \(\mathbb{R}\) is the real line with the metric \(dr^2\), and \(\varphi(x)\) is a \(C^2\) positive function on \(N\), then \(M_\varphi=(N\times\mathbb{R},\varphi^2(x)(\sigma+dr^2))\) is called a conformal product manifold, and if \(\Omega\) is a \(C^2\) bounded domain with compact closure \(\overline\Omega\) in \(N\), then \(Q_\varphi=\Omega\times\mathbb{R}\) in \(M_\varphi\) is called a conformal cone. If \(\psi(x)\) is a \(C^1\) function on \(\partial\Omega\) and \(\Gamma\) is its graph in \(\partial\Omega\times\mathbb{R}\), then the area minimizing problem in a conformal cone \(Q_\varphi\) is to find an \(n\)-integer multiplicity current in \(\overline Q_\varphi\) to realize
\[
\min\{\mathbb{M}(T);\ T\in\mathcal{G}\ \text{and}\ \partial T=\Gamma\}, \tag{\(*\)}
\]
where \(\mathbb{M}\) is the mass of integer multiplicity currents in \(M_\varphi\), and \(\mathcal{G}\) denotes the set of \(n\)-integer multiplicity currents with compact support in \(\overline{Q}_\varphi\), that is, for any \(T\in\mathcal{G}\), its support \(\text{spt}(T)\) is contained in \(\overline\Omega\times[a,b]\) for some finite numbers \(a < b\). If \(BV(W)\) is he set of all bounded variation functions on any open set \(W\), then a key concept for the study of the problem \((*)\) is an area functional in \(BV(W)\) defined as \( \mathfrak{F}_\varphi(u,W)=\sup\left\{\int_\Omega\{\varphi^n(x)h+u\,\text{div}(\varphi^n(x)X)\}\,d\,\text{vol}\right\} \) for \(h\in C_0(W)\), \(X\in T_0(W)\), and \(h^2+\langle X,X\rangle\le 1\), where \(d\text{vol}\) and \(\text{div}\) are the volume form and the divergence of \(N\), respectively, and \(C_0(W)\) and \(T_0(W)\) denote the set of smooth functions and vector fields with compact support in \(W\), respectively. If \(u\in C^1(W)\), then \(\mathfrak{F}_\varphi(u,W)\) is the area of the graph of \(u(x)\) in \(M_\varphi\). If \(\Omega\) is the \(C^2\) domain, \(\Omega'\) is a \(C^2\) domain in \(N\) satisfying \(\Omega\subset\!\subset\Omega'\), i.e., the closure of \(\Omega\) is a compact set in \(\Omega'\), and \(\psi(x)\in C^1(\Omega'\setminus\Omega)\), then the following minimizing problem:
\[
\min\{\mathfrak{F}_\varphi(v,\Omega');\ v(x)\in BV(\Omega'), v(x)=\psi(x)\ \text{on}\ \Omega'\setminus\Omega\} \tag{\(**\)}
\]
plays an important role to solve \((*)\). If \(\Sigma\) is a minimal graph of \(u(x)\) in \(M_\varphi\) over \(\Omega\) with \(C^1\) boundary \(\psi(x)\) on \(\partial\Omega\), then the Dirichlet problem is defined as
\[
\text{div}\!\left(\frac{D u}{\sqrt{1+|Du|^2}}\right)+n\left\langle D\log\varphi,\frac{D u}{\sqrt{1+|Du|^2}}\right\rangle=0 \tag{\(*\!*\!*\)}
\]
for \(x\in\Omega\), and \(u(x)=\psi(x)\) for \(x\in\partial\Omega\), where \(\psi(x)\) is a continuous function on \(\partial\Omega\) and div is the divergence of \(\Omega\). The key idea to solve \((*)\) is to establish the connection between the problem \((*)\), the area functional minimizing problem \((**)\), and the Dirichlet problem of minimal surface equations in \(M_\varphi\). If the mean curvature \(H\) of \(\partial\Omega\) satisfies \(H_{\partial\Omega}+n\langle\vec\gamma,D\log\varphi\rangle\ge 0\) on \(\partial\Omega\), where \(\vec\gamma\) is the outward normal vector of \(\partial\Omega\) and \(H_{\partial\Omega}= \text{div}(\vec{\gamma})\), then \(\Omega\) is called \(\varphi\)-mean convex. The authors show that if \(u(x)\) is the solution to the problem \((**)\), then \(T=\partial[\![U]\!]_{\overline{Q}_\varphi}\) solves the problem \((*)\) in \(M_\varphi\), where \(U\) is the subgraph of \(u(x)\) and \([\![U]\!]\) is the corresponding integer multiplicity current. As a direct application of this result is the Dirichlet problem of minimal surface equations in \(M_\varphi\). It is shown that if \(\Omega\) is \(\varphi\)-mean convex, then the Dirichlet problem \((*\!*\!*)\) with continuous boundary data has a unique solution in \(C^2(\Omega)\cap C(\overline\Omega)\). Finally, the authors consider the existence and uniqueness of local area minimizing integer multiplicity current in \(M_\varphi\) with infinity boundary \(\Gamma\) when \(\varphi(x)\) can be written as \(\varphi(d(x,\partial N))\) which goes to \(+\infty\) as \(d(x,\partial N)\to 0\) in \(N\), where \(N\) is a compact Riemannian manifold with \(C^2\) boundary and \(d\) is the distance function in \(N\). If \(N_r=\{x\in N;\ d(x,\partial N) > r\}\), then it is shown that if there is \(r_1\) such that for any \(r\in(0,r_1)\) \(N_r\) is \(\varphi\)-mean convex, then for any \(\psi(x)\in C(\partial N)\) and \(\Gamma=(x,\psi(x))\) there is a unique local area minimizing integer multiplicity current \(T\) with infinity boundary \(\Gamma\), and \(T\) is a minimal graph in \(M_\varphi\) over \(N\).On the unique solvability of the optimal starting control problem for the linearized equations of motion of a viscoelastic mediumhttps://zbmath.org/1472.490682021-11-25T18:46:10.358925Z"Artemov, M. A."https://zbmath.org/authors/?q=ai:artemov.mikhail-anatolevichSummary: We study an optimization problem for the linearized evolution equations of the Oldroyd model of motion of a viscoelastic medium. The equations are given in a bounded three-dimensional domain. The velocity distribution at the initial time is used as a control function. The objective functional is terminal. The existence of a unique optimal control is proved for a given set of admissible controls. A variational inequality characterizing the optimal control is derived.Geometric methods in physics XXXVIII. Workshop, Białowieża, Poland, June 30 -- July 6, 2019https://zbmath.org/1472.530062021-11-25T18:46:10.358925Z"Kielanowski, Piotr"https://zbmath.org/authors/?q=ai:kielanowski.piotr"Odzijewicz, Anatol"https://zbmath.org/authors/?q=ai:odzijewicz.anatol"Previato, Emma"https://zbmath.org/authors/?q=ai:previato.emmaPublisher's description: The book consists of articles based on the XXXVIII Białowieża Workshop on Geometric Methods in Physics, 2019. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. The works in this book, based on presentations given at the workshop, are previously unpublished, at the cutting edge of current research, typically grounded in geometry and analysis, with applications to classical and quantum physics.
For the past eight years, the Białowieża Workshops have been complemented by a School on Geometry and Physics, comprising series of advanced lectures for graduate students and early-career researchers. The extended abstracts of the five lecture series that were given in the eighth school are included. The unique character of the Workshop-and-School series draws on the venue, a famous historical, cultural and environmental site in the Białowieża forest, a UNESCO World Heritage Centre in the east of Poland: lectures are given in the Nature and Forest Museum and local traditions are interwoven with the scientific activities.
The articles of this volume will be reviewed individually. For the preceding workshop see [Zbl 1433.53003].
Indexed articles:
\textit{Arici, Francesca; Mesland, Bram}, Toeplitz extensions in noncommutative topology and mathematical physics, 3-29 [Zbl 07400843]
\textit{Beltiţă, Daniel; Odzijewicz, Anatol}, Standard groupoids of von Neumann algebras, 31-39 [Zbl 07400844]
\textit{Cotti, Giordano}, Quantum differential equations and helices, 41-65 [Zbl 07400845]
\textit{Dobrogowska, Alina; Mironov, Andrey E.}, Periodic one-point rank one commuting difference operators, 67-74 [Zbl 1472.39037]
\textit{Fehér, L.; Marshall, I.}, On the bi-Hamiltonian structure of the trigonometric spin Ruijsenaars-Sutherland hierarchy, 75-87 [Zbl 07400847]
\textit{Hara, Kentaro}, Hermitian-Einstein metrics from non-commutative \(U(1)\) solutions, 89-96 [Zbl 07400848]
\textit{Hounkonnou, Mahouton Norbert; Houndédji, Gbêvèwou Damien}, 2-hom-associative bialgebras and hom-left symmetric dialgebras, 97-115 [Zbl 07400849]
\textit{Crus y Cruz, S.; Gress, Z.; Jiménez-Macías, P.; Rosas-Ortiz, O.}, Laguerre-Gaussian wave propagation in parabolic media, 117-128 [Zbl 07400850]
\textit{Karmanova, Maria}, Maximal surfaces on two-step sub-Lorentzian structures, 129-141 [Zbl 07400851]
\textit{Lawson, Jimmie D.; Lim, Yongdo}, Following the trail of the operator geometric mean, 143-153 [Zbl 07400852]
\textit{Mandal, Ashis; Mishra, Satyendra Kumar}, On Hom-Lie-Rinehart algebras, 155-161 [Zbl 07400853]
\textit{Nakayashiki, Atsushi}, One step degeneration of trigonal curves and mixing of solitons and quasi-periodic solutions of the KP equation, 163-186 [Zbl 07400854]
\textit{Dobrokhotov, Sergei; Nazaikinskii, Vladimir}, Fock quantization of canonical transformations and semiclassical asymptotics for degenerate problems, 187-195 [Zbl 1472.81099]
\textit{Nieto, L. M.; Gadella, M.; Mateos-Guilarte, J.; Muñoz-Castañeda, J. M.; Romaniega, C.}, Some recent results on contact or point supported potentials, 197-219 [Zbl 07400856]
\textit{Orlov, Aleksandr Yu.}, 2D Yang-Mills theory and tau functions, 221-250 [Zbl 07400857]
\textit{Prorok, Dominik; Prykarpatski, Anatolij}, Many-particle Schrödinger type finitely factorized quantum Hamiltonian systems and their integrability, 251-270 [Zbl 07400858]
\textit{Quintana, C.; Jiménez-Macías, P.; Rosas-Ortiz, O.}, Quantum master equation for the time-periodic density operator of a single qubit coupled to a harmonic oscillator, 271-281 [Zbl 1472.81131]
\textit{Zelaya, Kevin; Cruz y Cruz, Sara; Rosas-Ortiz, Oscar}, On the construction of non-Hermitian Hamiltonians with all-real spectra through supersymmetric algorithms, 283-292 [Zbl 1472.81109]
\textit{Sontz, Stephen Bruce}, Toeplitz quantization of an analogue of the Manin plane, 293-301 [Zbl 07400861]
\textit{Przanowski, Maciej; Tosiek, Jaromir; Turrubiates, Francisco J.}, The Weyl-Wigner-Moyal formalism on a discrete phase space, 303-312 [Zbl 1472.81132]
\textit{Zheglov, Alexander}, Algebraic geometric properties of spectral surfaces of quantum integrable systems and their isospectral deformations, 313-331 [Zbl 1470.13043]
\textit{Domrin, A. V.}, Soliton equations and their holomorphic solutions, 335-343 [Zbl 07400864]
\textit{Goldin, Gerald A.}, Diffeomorphism groups in quantum theory and statistical physics, 345-350 [Zbl 1472.81114]
\textit{Rosas-Ortiz, Oscar}, Position-dependent mass systems: classical and quantum pictures, 351-361 [Zbl 07400866]
\textit{Slavnov, N. A.}, Introduction to the algebraic Bethe ansatz, 363-371 [Zbl 07400867]
\textit{Szymański, Wojciech}, Noncommutative fiber bundles, 373-379 [Zbl 07400868]On a class of almost Kenmotsu manifolds admitting an Einstein like structurehttps://zbmath.org/1472.530362021-11-25T18:46:10.358925Z"Dey, Dibakar"https://zbmath.org/authors/?q=ai:dey.dibakar"Majhi, Pradip"https://zbmath.org/authors/?q=ai:majhi.pradipSummary: In the present paper, we introduce the notion of \(*\)-gradient \(\rho \)-Einstein soliton on a class of almost Kenmotsu manifolds. It is shown that if a \((2n+1)\)-dimensional \((k,\mu )'\)-almost Kenmotsu manifold \(M\) admits \(*\)-gradient \(\rho \)-Einstein soliton with Einstein potential \(f\), then (1) the manifold \(M\) is locally isometric to \(\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n\), (2) the manifold \(M\) is \(*\)-Ricci flat and (3) the Einstein potential \(f\) is harmonic or satisfies a physical Poisson's equation. Finally, an illustrative example is presented.Results related to the Chern-Yamabe flowhttps://zbmath.org/1472.530412021-11-25T18:46:10.358925Z"Ho, Pak Tung"https://zbmath.org/authors/?q=ai:ho.pak-tungThe author studies, for a compact complex manifold \(X\) of complex dimension \(n\), endowed with a Hermitian metric \(\omega_0\), the Chern-Yamabe problem, i.e., to find a conformal metric of \(\omega_0\) such that its Chern scalar curvature is constant. In this paper, as a generalisation of the Chern-Yamabe problem, the author focuses on the problem of prescribing Chern scalar curvature. The main results, with proofs (using geometric flows related to the Chern-Yamabe flow), are:
\begin{itemize}
\item[1] the estimation of the first non-zero eigenvalue of Hodge-de Rham Laplacian of \((X, \omega_0)\),
\item[2] a proof of a version of conformal Schwarz lemma on \((X, \omega_0)\),
\item[3] a proof of the uniqueness of the Chern-Yamabe flow.
\end{itemize}A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponenthttps://zbmath.org/1472.530572021-11-25T18:46:10.358925Z"Bartsch, Thomas"https://zbmath.org/authors/?q=ai:bartsch.thomas.2|bartsch.thomas.1"Xu, Tian"https://zbmath.org/authors/?q=ai:xu.tianOn a compact spin manifold \((M^m,g)\), the authors study solutions of the nonlinear Dirac equation \(D\psi=\lambda\psi+f(|\psi|)\psi+|\psi|^{\frac{2}{m-1}}\psi\) where \(\lambda\in\mathbb{R}\) and \(f=o(s^{\frac{2}{m-1}})\) as \(s\to\infty\). Such an equation is called in the paper (NLD) and is the Euler-Lagrange equation associated to some functional \(\mathcal{L}_\lambda(\psi)\). The authors show in Theorem 2.1 that if \(f\) satisfies some conditions (called \(f_1\),\(f_2\) and \(f_3\) in the paper), then (NLD) has at least one energy solution \(\psi_\lambda\) for every \(\lambda>0\). Also the map \(\mathbb{R}^+\to \mathbb{R}^+; \lambda\to \mathcal{L}_\lambda(\psi_\lambda)\) is continuous and nonincreasing on each interval \([\lambda_k,\lambda_{k+1})\). On the other hand, if \(f\) satisfies some other conditions (called \(f_1\),\(f_4\) and \(f_5\) in the paper), then (NLD) has at least one energy solution \(\psi_\lambda\) for every \(\lambda\in \mathbb{R}\setminus \{\lambda_k:k\leq 0\}\). The map \(\mathbb{R}\setminus \{\lambda_k:k\leq 0\}\to \mathbb{R}^+; \lambda\to \mathcal{L}_\lambda(\psi_\lambda)\) is continuous and nonincreasing on each interval \([\lambda_{k-1},\lambda_{k})\), if \(k\geq 2\), respectively \((\lambda_{k-1},\lambda_{k})\), if \(k\leq 1\).Rigidity of surfaces with constant extrinsic curvature in Riemannian product spaceshttps://zbmath.org/1472.530732021-11-25T18:46:10.358925Z"dos Santos, Fábio R."https://zbmath.org/authors/?q=ai:dos-santos.fabio-reisSummary: The present paper deals with complete surfaces having constant extrinsic curvature in a Riemannian product space \(M^2(c)\times\mathbb{R}\), where \(M^2(c)\) is a space form with constant sectional curvature \(c\in\{-1,1\}\). In such setting, we find a Simons-type formula for Cheng-Yau's operator which is used to prove that such surfaces are isometric to a cylinder \(\mathbb{H}^1\times\mathbb{R}\), when \(c=-1\) or isometric to a slice \(\mathbb{S}^2\times\{t_0\}\) for some \(t_0\in\mathbb{R}\) when \(c=1\). Finally, we extend the result, when \(c=-1\), for the Weingarten linear case.Weak convergence of branched conformal immersions with uniformly bounded areas and Willmore energieshttps://zbmath.org/1472.530772021-11-25T18:46:10.358925Z"Wei, Guodong"https://zbmath.org/authors/?q=ai:wei.guodongThe author extends \textit{F. Hélein}'s convergence theorem [Harmonic maps, conservation laws and moving frames. Transl. from the French. Cambridge: Cambridge University Press (2002; Zbl 1010.58010)] to a sequence of rescaled branched conformal immersions. He applies this result to study the blow-up behavior of a sequence of branched conformal immersions of a closed Riemann surface into \(\mathbb{R}^n\) whose areas and Wilmore energies are uniformly bounded. In this situation he gives a bubble tree construction and proves an integral identity for the Gauss curvature of the limit.CR-harmonic mapshttps://zbmath.org/1472.530782021-11-25T18:46:10.358925Z"Dietrich, Gautier"https://zbmath.org/authors/?q=ai:dietrich.gautierSummary: We develop the notion of renormalized energy in Cauchy-Riemann (CR) geometry for maps from a strictly pseudoconvex pseudo-Hermitian manifold to a Riemannian manifold. This energy is a CR invariant functional whose critical points, which we call CR-harmonic maps, satisfy a CR covariant partial differential equation. The corresponding operator coincides on functions with the CR Paneitz operator.A finiteness theorem for holonomic DQ-modules on Poisson manifoldshttps://zbmath.org/1472.530972021-11-25T18:46:10.358925Z"Kashiwara, Masaki"https://zbmath.org/authors/?q=ai:kashiwara.masaki"Schapira, Pierre"https://zbmath.org/authors/?q=ai:schapira.pierreSummary: On a complex symplectic manifold, we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification, (b) in the algebraic case. This extends our previous result in which the symplectic manifold was compact. The main tool is a finiteness theorem for \(\mathbb{R}\)-constructible sheaves on a real analytic manifold in a nonproper situation.Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold \(M^2\times\mathbb{R}\)https://zbmath.org/1472.530982021-11-25T18:46:10.358925Z"Chen, Li"https://zbmath.org/authors/?q=ai:chen.li.2|chen.li.3|chen.li.4|chen.li.1|chen.li.6|chen.li.7|chen.li.5"Hu, Dan-Dan"https://zbmath.org/authors/?q=ai:hu.dandan"Mao, Jing"https://zbmath.org/authors/?q=ai:mao.jing"Xiang, Ni"https://zbmath.org/authors/?q=ai:xiang.niSummary: In this paper, for the Lorentz manifold \(M^2\times\mathbb{R}\) with \(M^2\) a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in \(M^2\), which are evolving by the non-parametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.Local existence and uniqueness of skew mean curvature flowhttps://zbmath.org/1472.531002021-11-25T18:46:10.358925Z"Song, Chong"https://zbmath.org/authors/?q=ai:song.chongSummary: The Skew Mean Curvature Flow (SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general-dimensional SMCF in Euclidean spaces.Type II ancient compact solutions to the Yamabe flowhttps://zbmath.org/1472.531022021-11-25T18:46:10.358925Z"Daskalopoulos, Panagiota"https://zbmath.org/authors/?q=ai:daskalopoulos.panagiota"del Pino, Manuel"https://zbmath.org/authors/?q=ai:del-pino.manuel-a"Sesum, Natasa"https://zbmath.org/authors/?q=ai:sesum.natasaThe authors construct new type II ancient compact solutions to the Yamabe flow. These solutions are rotationally symmetric and converge to a tower of two spheres as \(t\to -\infty\). The ancient solutions to the Yamabe flow are constructed by gluing two exact solutions to the rescaled equations, that is spheres, with narrow cylindrical necks. They use perturbation theory via fixed points arguments based on sharp estimates on ancient solutions of the approximated linear equation and a careful estimation of the error terms. This result can be generalized to the gluing of \(k\) spheres for any \(k\ge 2\) in such a way that the configuration of radii of the spheres was driven by a first-order Toda system as \(t\to -\infty\).Complete Ricci solitons via estimates on the soliton potentialhttps://zbmath.org/1472.531042021-11-25T18:46:10.358925Z"Wink, Matthias"https://zbmath.org/authors/?q=ai:wink.matthiasSummary: In this paper, a growth estimate on the soliton potential is shown for a large class of cohomogeneity one manifolds. This is used to construct continuous families of complete steady and expanding Ricci solitons in the setups of
\textit{H. Lü} et al. [Phys. Lett., B 593, No. 1--4, 218--226 (2004; Zbl 1247.53054)]
and
\textit{A. S. Dancer} and \textit{M. Y. Wang} [Ann. Global Anal. Geom. 39, No. 3, 259--292 (2011; Zbl 1215.53040)].
It also provides a different approach to the two summands system
[the author, ``Cohomogeneity one Ricci solitons from Hopf fibrations'', Preprint, \url{arXiv:1706.09712}]
that applies to all known geometric examples.Stability of the conical Kähler-Ricci flows on Fano manifoldshttps://zbmath.org/1472.531062021-11-25T18:46:10.358925Z"Liu, Jiawei"https://zbmath.org/authors/?q=ai:liu.jiawei"Zhang, Xi"https://zbmath.org/authors/?q=ai:zhang.xiSummary: In this paper, we study stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle \(2\pi\beta\) along the divisor, then for any \(\beta'\) sufficiently close to \(\beta\), the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle \(2\pi\beta'\) along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence. As applications, we give parabolic proofs of Donaldson's openness theorem and his conjecture for the existence of conical Kähler-Einstein metrics with positive Ricci curvatures.Gauss curvature flow with an obstaclehttps://zbmath.org/1472.531072021-11-25T18:46:10.358925Z"Lee, Ki-Ahm"https://zbmath.org/authors/?q=ai:lee.ki-ahm"Lee, Taehun"https://zbmath.org/authors/?q=ai:lee.taehunSummary: We consider the obstacle problem for the Gauss curvature flow with an exponent \(\alpha\). Under the assumption that both the obstacle and the initial hypersurface are strictly convex closed hypersurfaces and that the obstacle is enclosed by the initial hypersurface, uniform estimates are obtained for several curvatures via a penalty method. We also prove that when the hypersurface is two dimensional with \(0<\alpha \le 1\), the solution of the Gauss curvature flow with an obstacle exists for all time with bounded principal curvatures \(\{\lambda_i\}\), where the upper bound is uniform, and the lower bound depends on the distance from the free boundary. Moreover, we show that there exists a finite time \(T_*\) after which the solution becomes stationary in the same shape as the obstacle.Ephemeral persistence modules and distance comparisonhttps://zbmath.org/1472.550032021-11-25T18:46:10.358925Z"Berkouk, Nicolas"https://zbmath.org/authors/?q=ai:berkouk.nicolas"Petit, François"https://zbmath.org/authors/?q=ai:petit.francoisThe paper studies multiparameter persistence modules by a sheaf-theoretic approach. Extracting appropriate features from multiparameter persistence modules is difficult and many approaches have been proposed, for example, using polynomial rings and quiver representations. Another approach using sheaf theory is proposed by [\textit{J. Curry}, Sheaves, cosheaves and applications. (Ph.D. thesis) University of Pennsylvania (2014), \url{arXiv:1303.3255}] and [\textit{M. Kashiwara} and \textit{P. Schapira}, J. Appl. Comput. Topol. 2, No. 1-2, 83--113 (2018; Zbl 1423.55013)]. The current paper reveals more about the significance of sheaf theory, in particular, \(\gamma\)-sheaves, for studying persistent homology.
One of the aims of the paper is to compare the two approaches of Curry and Kashiwara-Schapira, where the former uses Alexandrov topology and the latter uses \(\gamma\)-topology, where \(\gamma\) is a closed convex proper cone in a finite-dimensional real vector space \(\mathbb{V}\). The authors consider a morphism of sites between Alexandrov and \(\gamma\)-topology spaces \(\beta \colon \mathbb{V}_\mathfrak{a} \to \mathbb{V}_\gamma\), which induces a functor between the categories of sheaves of \(\mathbf{k}\)-vector spaces \(\beta_* \colon \mathrm{Mod}(\mathbf{k}_{\mathbb{V}_\mathfrak{a}}) \to \mathrm{Mod}(\mathbf{k}_{\mathbb{V}_\gamma})\). The category of ephemeral modules \(\mathrm{Eph}(\mathbf{k}_{\mathbb{V}_\mathfrak{a}})\) is now defined as the kernel of the functor. In this way, the notion of ephemeral modules can be defined in arbitrary dimension and coincides with the definition of [\textit{F. Chazal} et al., The structure and stability of persistence modules. Cham: Springer (2016; Zbl 1362.55002)] in the one-dimensional case. An equivalence between \(\mathrm{Mod}(\mathbf{k}_{\mathbb{V}_\mathfrak{a}})/\mathrm{Eph}(\mathbf{k}_{\mathbb{V}_\mathfrak{a}})\) and \(\mathrm{Mod}(\mathbf{k}_{\mathbb{V}_\gamma})\) is also obtained, which shows the equivalence between the observable category and \(\gamma\)-sheaves. The authors extend the results to the derived setting.
The latter half of the paper studies the distances on the two categories. For a fixed vector \(v\), both categories admit interleaving distances to the \(v\)-direction. The authors show an isometry theorem on these metrics. Finally, the paper investigates the relation between the convolution distance defined by Kashiwara-Schapira and the interleaving distance on the category of sheaves on \(\mathbb{V}_\gamma\). The convolution distance depends on the norm on \(\mathbb{V}\), whereas the interleaving distance depends on \(v\). The authors introduce a preferred norm and prove an isometry theorem on these metrics under a mild assumption.Fractional moments of the stochastic heat equationhttps://zbmath.org/1472.600512021-11-25T18:46:10.358925Z"Das, Sayan"https://zbmath.org/authors/?q=ai:das.sayan-kumar"Tsai, Li-Cheng"https://zbmath.org/authors/?q=ai:tsai.li-chengSummary: Consider the solution \(\mathcal{Z}(t,x)\) of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data \(Z(0,x)=\delta(x)\). For any real \(p>0\), we obtained detailed estimates of the \(p\)th moment of \(e^{t/12}\mathcal{Z}(2t,0)\), as \(t\to\infty\), and from these estimates establish the one-point upper-tail large deviation principle of the Kardar-Parisi-Zhang equation. The deviations have speed \(t\) and rate function \(\Phi_+(y)=\frac{4}{3}y^{3/2}\). Our result confirms the existing physics predictions [\textit{P. Le Doussal} et al., ``Large deviations for the height in 1D Kardar-Parisi-Zhang growth at late times'', Europhys. Lett. 113, No. 6, Article ID 60004, 6 p. (2016; \url{doi:10.1209/0295-5075/113/60004})] and also [\textit{A. Kamenev} et al., ``Short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation: starting from a parabola'', Phys. Rev. E 94, No. 3, Article ID 032108, 9 p. (2016; \url{doi:10.1103/PhysRevE.94.032108})].Uniform large deviation principles for Banach space valued stochastic evolution equationshttps://zbmath.org/1472.600522021-11-25T18:46:10.358925Z"Salins, Michael"https://zbmath.org/authors/?q=ai:salins.michael"Budhiraja, Amarjit"https://zbmath.org/authors/?q=ai:budhiraja.amarjit-s"Dupuis, Paul"https://zbmath.org/authors/?q=ai:dupuis.paul-gSummary: We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDEs) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform LDP over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite-dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak-\( \star\) compactness of closed bounded sets in the double dual space. We prove that a modified version of our SDE satisfies a uniform Laplace principle over weak-\( \star\) compact sets and consequently a uniform over bounded sets LDP. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite-dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and two-dimensional stochastic Navier-Stokes equations with multiplicative noise.Global \(C^1\) regularity of the value function in optimal stopping problemshttps://zbmath.org/1472.600732021-11-25T18:46:10.358925Z"de Angelis, Tiziano"https://zbmath.org/authors/?q=ai:de-angelis.tiziano"Peskir, Goran"https://zbmath.org/authors/?q=ai:peskir.goranSummary: We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic case of the boundary value problem under the sole hypothesis of probabilistic regularity of the optimal stopping boundary, thus improving upon known analytic results in the PDE literature, and establishing the fact for the first time in the case of integro-differential equations. The method of proof is purely probabilistic and conceptually simple. Examples of application include the first known probabilistic proof of the fact that the time derivative of the value function in the American put problem is continuous across the optimal stopping boundary.Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumpshttps://zbmath.org/1472.600832021-11-25T18:46:10.358925Z"Friesen, Martin"https://zbmath.org/authors/?q=ai:friesen.martin"Jin, Peng"https://zbmath.org/authors/?q=ai:jin.peng"Rüdiger, Barbara"https://zbmath.org/authors/?q=ai:rudiger.barbaraSummary: We study existence and Besov regularity of densities for solutions to stochastic differential equations with Hölder continuous coefficients driven by a \(d\)-dimensional Lévy process \(Z=(Z(t))_{t\geq 0}\), where, for \(t>0\), the density function \(f_t\) of \(Z(t)\) exists and satisfies, for some \((\alpha_i)_{i=1,\dots,d}\subset (0,2)\) and \(C>0\),
\[
\limsup\limits_{t\to 0}t^{1/\alpha_i}\int_{\mathbb{R}^d}|f_t(z+e_ih)-f_t(z)|dz\leq C|h|,\quad h\in\mathbb{R},i=1,\dots,d.
\]
Here \(e_1,\dots,e_d\) denote the canonical basis vectors in \(\mathbb{R}^d\). The latter condition covers anisotropic \((\alpha_1,\dots,\alpha_d)\)-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from [\textit{A. Debussche} and \textit{N. Fournier}, J. Funct. Anal. 264, No. 8, 1757--1778 (2013; Zbl 1272.60032)].The Osgood condition for stochastic partial differential equationshttps://zbmath.org/1472.601002021-11-25T18:46:10.358925Z"Foondun, Mohammud"https://zbmath.org/authors/?q=ai:foondun.mohammud"Nualart, Eulalia"https://zbmath.org/authors/?q=ai:nualart.eulaliaSummary: We study the following equation
\[
\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x)+b\bigl(u(t,x)\bigr)+\sigma \dot{W}(t,x),\quad t>0,
\]
where \(\sigma\) is a positive constant and \(\dot{W}\) is a space-time white noise. The initial condition \(u(0,x)=u_0(x)\) is assumed to be a nonnegative and continuous function. We first study the problem on \([0,1]\) with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of \textit{J. F. Bonder} and \textit{P. Groisman} [Physica D 238, No. 2, 209--215 (2009; Zbl 1173.35543)], our first result shows that the solution blows up in finite time if and only if for some \(a>0\),
\[
\int_a^{\infty }\frac{1}{b(s)}\,\mathrm{d}s<\infty,
\]
which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in \(\mathbf{R}^d \).Paracontrolled quasi-geostrophic equation with space-time white noisehttps://zbmath.org/1472.601022021-11-25T18:46:10.358925Z"Inahama, Yuzuru"https://zbmath.org/authors/?q=ai:inahama.yuzuru"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiroThe authors consider the stochastic dissipative quasi-geostrophic equations
\[
\partial_tu^\varepsilon=-(-\Delta)^{\theta/2}u^\varepsilon+R\perp u^\varepsilon\cdot\nabla u^\varepsilon+\xi^\varepsilon,\qquad u^\varepsilon(0)=u_0
\]
on the two-dimensional torus \(\mathbb T^2\), where \(\varepsilon\in(0,1)\), \(\theta\in(7/4,2]\), \(R=(R_1,R_2)\) with \(R_j\) being the \(j\)-th Riesz transform on \(\mathbb T^2\), \(\xi\) is an additive space-time white noise on \(\mathbb T^2\) and \(\xi^\varepsilon\) for \(\varepsilon>0\) are mollifications of \(\xi \). If \(\kappa>\kappa_\theta\) for an explicitly defined threshold \(\kappa_\theta\) and \(u_0\in\mathcal C^\kappa\) where \(\mathcal C^\kappa\) is the Besov-Hölder space \(B^\kappa_{\infty,\infty}\) on \(\mathbb T^2\) then the unique solutions \(u^\varepsilon\) exist on \([0,T^\varepsilon_*]\) where \(T^\varepsilon_*\) are positive random times, \(T^\varepsilon_*\) converge in probability to a positive random time \(T_*\) as \(\varepsilon\to 0+\), and there exists a \(C^\kappa\)-valued process \(u\) on \([0,T_*)\) such that
\[
\lim_{\varepsilon\to 0+}\left(\sup_{s\in[0,T^\varepsilon_*\land\frac{T_*}{2}]}\|u^\varepsilon_s-u_s\|_{\mathcal C^{\kappa}}\right)=0
\]
in probability. Moreover, \(u\) is independent of the mollification.Large time asymptotic properties of the stochastic heat equationhttps://zbmath.org/1472.601032021-11-25T18:46:10.358925Z"Kohatsu-Higa, Arturo"https://zbmath.org/authors/?q=ai:kohatsu-higa.arturo"Nualart, David"https://zbmath.org/authors/?q=ai:nualart.davidIn this paper, the authors studied the large time asymptotic behavior of the stochastic heat equation. In particular, the authors discussed an extension of these results in the case of space averages on an interval \([-R, R]\) and both \(R\) and \(t\) tend to infinity. Further, the case of nonlinear equations is also discussed.Locally robust random attractors in stochastic non-autonomous magneto-hydrodynamicshttps://zbmath.org/1472.601042021-11-25T18:46:10.358925Z"Li, Fuzhi"https://zbmath.org/authors/?q=ai:li.fuzhi"Yangrong, Li"https://zbmath.org/authors/?q=ai:yangrong.liThis paper shows the local robustness result of random attractors (towards a deterministic attractor), which generalizes some related results in the literature. The main results are illustrated in the content of stochastic non-autonomous magneto-hydrodynamics (MHD) equations. By using joint convergence of the cocycles, collective local compactness and deterministic recurrence of the random attractors, the authors prove that the family of pullback random attractors is locally uniform convergent to the pullback attractor of the deterministic MHD equation when the density of random noise tends to 0.Representation of the solution of Goursat problem for second order linear stochastic hyperbolic differential equationshttps://zbmath.org/1472.601052021-11-25T18:46:10.358925Z"Mansimov, Kamil' Baĭramali"https://zbmath.org/authors/?q=ai:mansimov.kamil-bairamali-oglu"Mastaliev, Rashad Ogtaĭ"https://zbmath.org/authors/?q=ai:mastaliev.rashad-ogtai-oglySummary: The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noisehttps://zbmath.org/1472.601062021-11-25T18:46:10.358925Z"Wang, Xiaohu"https://zbmath.org/authors/?q=ai:wang.xiaohu"Li, Dingshi"https://zbmath.org/authors/?q=ai:li.dingshi"Shen, Jun"https://zbmath.org/authors/?q=ai:shen.junThe authors investigate the asymptotic behavior of the solutions of the stochastic wave equation driven by an additive white noise on unbounded domains and its Wong-Zakai approximation.
The main idea of Wong-Zakai approximation is to use deterministic differential equations to approximate the solutions of the stochastic differential equations.
In the article, Brownian motion is approximated using the Euler approximation. The authors prove the existence and uniqueness of tempered pullback attractors for stochastic wave equation and its Wong-Zakai approximation. Then, they show that the attractor of the Wong-Zakai approximate equation converges to the one of the stochastic wave equation driven by additive noise as the correlation time of noise approaches zero.Extensions and solutions for nonlinear diffusion equations and random walkshttps://zbmath.org/1472.601092021-11-25T18:46:10.358925Z"Lenzi, E. K."https://zbmath.org/authors/?q=ai:kaminski-lenzi.ervin"Lenzi, M. K."https://zbmath.org/authors/?q=ai:lenzi.marcelo-k"Ribeiro, H. V."https://zbmath.org/authors/?q=ai:ribeiro.haroldo-v"Evangelista, L. R."https://zbmath.org/authors/?q=ai:evangelista.luiz-robertoSummary: We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify that framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.Non-asymptotic Gaussian estimates for the recursive approximation of the invariant distribution of a diffusionhttps://zbmath.org/1472.601122021-11-25T18:46:10.358925Z"Honoré, I."https://zbmath.org/authors/?q=ai:honore.igor"Menozzi, S."https://zbmath.org/authors/?q=ai:menozzi.stephane"Pagès, G."https://zbmath.org/authors/?q=ai:pages.gilles|pages.gaelSummary: We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant distribution \(\nu\) of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions \(f\) such that \(f-\nu (f)\) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when some suitable squared-norms of the diffusion coefficient also belong to this class. We apply these estimates to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.Target competition for resources under multiple search-and-capture events with stochastic resettinghttps://zbmath.org/1472.601492021-11-25T18:46:10.358925Z"Bressloff, P. C."https://zbmath.org/authors/?q=ai:bressloff.paul-cSummary: We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed location \(x_{}r\) at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to \(x_{}r\), where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain with \(N\) small interior targets. We show that slow resetting increases the total number of resources \(M_{tot}\) across all targets provided that \(\sum_{j = 1}^N G( \text{x}_r, \text{x}_j) < 0\), where \(G\) is the Neumann Green's function and \(x_{}j\) is the location of the \(j\)-th target. This implies that \(M_{tot}\) can be optimized by varying \(r\). We also show that the \(k\)-th target has a competitive advantage if \(\sum_{j = 1}^N G( \text{x}_r, \text{x}_j) > N G( \text{x}_r, \text{x}_k)\).Spectral gap in mean-field \({\mathcal{O}}(n)\)-modelhttps://zbmath.org/1472.601502021-11-25T18:46:10.358925Z"Becker, Simon"https://zbmath.org/authors/?q=ai:becker.simon"Menegaki, Angeliki"https://zbmath.org/authors/?q=ai:menegaki.angelikiSummary: We study the dependence of the spectral gap for the generator of the Ginzburg-Landau dynamics for all \(\mathcal O(n)\)-\textit{models} with mean-field interaction and magnetic field, below and at the critical temperature on the number \(N\) of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schrödinger operator.Learning on dynamic statistical manifoldshttps://zbmath.org/1472.620112021-11-25T18:46:10.358925Z"Boso, F."https://zbmath.org/authors/?q=ai:boso.francesca"Tartakovsky, D. M."https://zbmath.org/authors/?q=ai:tartakovsky.daniel-mSummary: Hyperbolic balance laws with uncertain (random) parameters and inputs are ubiquitous in science and engineering. Quantification of uncertainty in predictions derived from such laws, and reduction of predictive uncertainty via data assimilation, remain an open challenge. That is due to nonlinearity of governing equations, whose solutions are highly non-Gaussian and often discontinuous. To ameliorate these issues in a computationally efficient way, we use the method of distributions, which here takes the form of a deterministic equation for spatio-temporal evolution of the cumulative distribution function (CDF) of the random system state, as a means of forward uncertainty propagation. Uncertainty reduction is achieved by recasting the standard loss function, i.e. discrepancy between observations and model predictions, in distributional terms. This step exploits the equivalence between minimization of the square error discrepancy and the Kullback-Leibler divergence. The loss function is regularized by adding a Lagrangian constraint enforcing fulfilment of the CDF equation. Minimization is performed sequentially, progressively updating the parameters of the CDF equation as more measurements are assimilated.Consistency of empirical Bayes and kernel flow for hierarchical parameter estimationhttps://zbmath.org/1472.620122021-11-25T18:46:10.358925Z"Chen, Yifan"https://zbmath.org/authors/?q=ai:chen.yifan"Owhadi, Houman"https://zbmath.org/authors/?q=ai:owhadi.houman"Stuart, Andrew M."https://zbmath.org/authors/?q=ai:stuart.andrew-mSummary: Gaussian process regression has proven very powerful in statistics, machine learning and inverse problems. A crucial aspect of the success of this methodology, in a wide range of applications to complex and real-world problems, is hierarchical modeling and learning of hyperparameters. The purpose of this paper is to study two paradigms of learning hierarchical parameters: one is from the probabilistic Bayesian perspective, in particular, the empirical Bayes approach that has been largely used in Bayesian statistics; the other is from the deterministic and approximation theoretic view, and in particular the kernel flow algorithm that was proposed recently in the machine learning literature. Analysis of their consistency in the large data limit, as well as explicit identification of their implicit bias in parameter learning, are established in this paper for a Matérn-like model on the torus. A particular technical challenge we overcome is the learning of the regularity parameter in the Matérn-like field, for which consistency results have been very scarce in the spatial statistics literature. Moreover, we conduct extensive numerical experiments beyond the Matérn-like model, comparing the two algorithms further. These experiments demonstrate learning of other hierarchical parameters, such as amplitude and lengthscale; they also illustrate the setting of model misspecification in which the kernel flow approach could show superior performance to the more traditional empirical Bayes approach.Spectral convergence of graph Laplacian and heat kernel reconstruction in \(L^\infty\) from random sampleshttps://zbmath.org/1472.621782021-11-25T18:46:10.358925Z"Dunson, David B."https://zbmath.org/authors/?q=ai:dunson.david-b"Wu, Hau-Tieng"https://zbmath.org/authors/?q=ai:wu.hau-tieng"Wu, Nan"https://zbmath.org/authors/?q=ai:wu.nanSummary: In the manifold setting, we provide a series of spectral convergence results quantifying how the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator in the \(L^\infty\) sense. Based on these results, convergence of the proposed heat kernel approximation algorithm, as well as the convergence rate, to the exact heat kernel is guaranteed. To our knowledge, this is the first work exploring the spectral convergence in the \(L^\infty\) sense and providing a numerical heat kernel reconstruction from the point cloud with theoretical guarantees.Moving boundary PDE analysis. Biomedical applications in R.https://zbmath.org/1472.650022021-11-25T18:46:10.358925Z"Schiesser, William E."https://zbmath.org/authors/?q=ai:schiesser.william-eFrom the preface: This book is directed to the numerical integration (solution) of systems of partial differential equations (PDEs) for which the boundary conditions move in space. In applications, the physical boundaries move as the solution evolves in time.
Moving boundary PDEs (MBPDEs) are an important class of mathematical models with a spectrum of applications. In this book, the focus is on two biomedical applications. The first example (Chapter 4) pertains to the boundary of a tumor that moves outward as the tumor grows. The second example (Chapter 5) pertains to the movement of the inner wall of an artery as plaque forms during atherosclerosis; if the plaque formation continues long enough, the arterial bloodstream is seriously restricted, a precursor to stroke and myocardial infarction (heart attack).
Chapters 1 through 3 discuss a numerical algorithm (computational method) for MFPDEs that to the author's knowledge is an original contribution. The algorithm is implemented in computer routines that are coded in R, a quality open-source scientific programming system. The routines are validated with a series of special-case MFPDE test problems.
The routines are available from a download link so that the reader/analyst/researcher can execute the test problems and example applications discussed in the book without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the reported test problems and applications, such as changes in the MBPDE parameters (constants) and form of the model equations. Finally, the generic routines can be applied to new models, and the associated R routines can be readily executed on modest computers.Model reduction for kinetic equations: moment approximations and hierarchical approximate proper orthogonal decompositionhttps://zbmath.org/1472.650042021-11-25T18:46:10.358925Z"Leibner, Tobias"https://zbmath.org/authors/?q=ai:leibner.tobias(no abstract)Iterative properties of solution for a general singular \(n\)-Hessian equation with decreasing nonlinearityhttps://zbmath.org/1472.650662021-11-25T18:46:10.358925Z"Zhang, Xinguang"https://zbmath.org/authors/?q=ai:zhang.xinguang"Jiang, Jiqiang"https://zbmath.org/authors/?q=ai:jiang.jiqiang"Wu, Yonghong"https://zbmath.org/authors/?q=ai:wu.yonghong.1"Wiwatanapataphee, Benchawan"https://zbmath.org/authors/?q=ai:wiwatanapataphee.benchawanThis paper deals with the iterative properties of solutions for a class of general singular \(n\)-Hessian equations with Dirichlet boundary condition of the form
\[
(-1)^{n}S_{n}^{\frac{1}{n}}(\mu (D^{2}u))= g(|x|, u),\text{ in }\Omega\subset R^{N},\ (n< N <2n)\tag{1}
\]
\[
u=0\text{ on }\partial \Omega,\tag{2}
\]
where \(\Omega\) is a unit ball, \(g:\Omega/\{0\}\times (0,\infty)\rightarrow [0,+\infty)\) is continuous, \(S_{n}(\mu(D^{2}u))\) is a \(n\)-Hessian operator defined by
\[
S_{n}(\mu (D^{2}u))=\sum_{1\leq i_{1}<i_{2}<\dots<i_{n}\leq N}\mu_{i_{1}}\cdot\mu_{i_{2}}\cdots\mu_{i_{n}},\ n=1,2,\dots,N,
\]
where \(\mu_{1},\mu_{2},\dots,\mu_{N}\) are eigenvalues of the Hessian matrix \(D^{2}(u)\) and \(\mu(D^{2}(u)) = (\mu_{1},\mu_{2},\dots,\mu_{N})\) is the vector of the eigenvalues of \(D^{2}(u)\).
In this paper, the authors have introduced a double iterative technique to establish the criterion of the existence for unique solution of the equations (1)-(2). They study the convergence properties of the solution, error estimation and the convergence rate between iterative value and exact solution. It may be noted that the nonlinear function involved in (1) is decreasing and allows a stronger singularity at some points of the time or space variables. Thus this study is different from earlier ones.
An example is given to illustrate their main result.Non-standard finite difference based numerical method for viscous Burgers' equationhttps://zbmath.org/1472.650982021-11-25T18:46:10.358925Z"Clemence-Mkhope, D. P."https://zbmath.org/authors/?q=ai:clemence-mkhope.d-p"Rabeeb Ali, V. P."https://zbmath.org/authors/?q=ai:rabeeb-ali.v-p"Awasthi, Ashish"https://zbmath.org/authors/?q=ai:awasthi.ashishSummary: Simple and accurate non-standard finite difference (NSFD) schemes are proposed for solving the viscous Burgers' equation with and without Cole-Hopf transformation. The stability and positivity of the schemes are discussed in detail, and the schemes are very user friendly in implementation. For some classic examples for the viscous Burgers' equation, simulation outcomes are validated through comparison with exact solutions for small and large values of kinematic viscosity. Numerical results from the NSFD schemes and corresponding standard finite difference methods are tabulated and compared with results from some existing methods. The proposed schemes give relatively accurate results, with comparatively less observed absolute errors.Convergence of time-splitting approximations for degenerate convection-diffusion equations with a random sourcehttps://zbmath.org/1472.650992021-11-25T18:46:10.358925Z"Díaz-Adame, Roberto"https://zbmath.org/authors/?q=ai:diaz-adame.roberto"Jerez, Silvia"https://zbmath.org/authors/?q=ai:jerez.silviaIn this paper authors propose a time-splitting method for degenerate convectionś-diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence in \(L_{\mathrm{loc}}^p\) of the time-splitting operator scheme to the unique weak entropy solution is proven. Moreover, we analyze the performance of the splitting approximation by computing its convergence rate and showing numerical simulations for some benchmark examples, including a luid low application in porous media.Efficient modified techniques of invariant energy quadratization approach for gradient flowshttps://zbmath.org/1472.651002021-11-25T18:46:10.358925Z"Liu, Zhengguang"https://zbmath.org/authors/?q=ai:liu.zhengguang"Li, Xiaoli"https://zbmath.org/authors/?q=ai:li.xiaoliSummary: Two novel and efficient modified techniques based on recently developed stabilized invariant energy quadratization (IEQ) approach to deal with nonlinear terms in gradient flows are proposed in this paper. We proved the unconditional energy stability for a class of gradient flows and their semi-discrete schemes carefully and rigorously. One of the contributions for this approach is that we succeeded in finding suitable positive preserving functions in square root and do not need to add a positive constant which cannot be fixed before computing. Secondly, all nonlinear terms can be treated semi-explicitly, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.The impact of Chebyshev collocation method on solutions of fractional advection-diffusion equationhttps://zbmath.org/1472.651012021-11-25T18:46:10.358925Z"Mesgarani, H."https://zbmath.org/authors/?q=ai:mesgarani.hamid"Rashidnina, J."https://zbmath.org/authors/?q=ai:rashidnina.j"Aghdam, Y. Esmaeelzade"https://zbmath.org/authors/?q=ai:aghdam.y-esmaeelzade"Nikan, O."https://zbmath.org/authors/?q=ai:nikan.omidSummary: The present paper is primarily aimed at obtaining the numerical solution of space fractional advection-diffusion equation including two fractional space derivatives of order. At the first stage, a difference approach with the second-order accuracy is formulated to obtain a semi-discrete scheme. Unconditional stability and convergence analysis has been analyzed. At the second stage, the spatial discretization is accomplished by means of the second kind shifted Chebyshev polynomials. Two examples are investigated, and numerical results are reported to confirm the theoretical results.Mean-square convergence of projection-difference method with the scheme of Crank-Nicolson in time for the approximate solution of parabolic equation with an integral conditionon the solutionhttps://zbmath.org/1472.651022021-11-25T18:46:10.358925Z"Smagin, V. V."https://zbmath.org/authors/?q=ai:smagin.viktor-vSummary: In Hilbert space, an abstract linear parabolic equation in a variational form with a symmetric operator and a nonlocal integral condition on the solution is solved approximately by the projection-difference method using the Krank-Nicholson scheme, which is a second-order approximation scheme. Approximation of the problem with respect to spatial variables is constructed from an arbitrary system of finite-dimensional subspaces and is oriented to the finite element method. The root-mean-square estimates of the errors of approximate solutions and the convergence of approximate solutions to the exact solution are obtained. The rate of this convergence is established with the second order in time, in addition, this velocity is exact in the order of approximation with respect to the spatial variables.The pointwise error estimates of two energy-preserving fourth-order compact schemes for viscous Burgers' equationhttps://zbmath.org/1472.651032021-11-25T18:46:10.358925Z"Wang, Xuping"https://zbmath.org/authors/?q=ai:wang.xuping"Zhang, Qifeng"https://zbmath.org/authors/?q=ai:zhang.qifeng"Sun, Zhi-zhong"https://zbmath.org/authors/?q=ai:sun.zhizhongSummary: A novel fourth-order three-point compact operator for the nonlinear convection term \(uu_x\) is provided in this paper. The operator makes the numerical analysis of higher-order difference schemes become possible for a wide class of nonlinear evolutionary equations under the unified framework. We take the classical viscous Burgers' equation as an example and establish a new conservative fourth-order implicit compact difference scheme based on the method of order reduction. A detailed theoretical analysis is carried out by the discrete energy argument and mathematical induction. It is rigorously proved that the difference scheme is conservative, uniquely solvable, stable, and unconditionally convergent in discrete \(L^{\infty}\)-norm. The convergence order is two in time and four in space, respectively. Furthermore, we derive a three-level linearized compact difference scheme for viscous Burgers' equation based on the proposed operator. All numerically theoretical results similar to that of the nonlinear numerical scheme are inherited completely; meanwhile, it is more time saving. Applying the compact operator to other more complex and higher-order nonlinear evolutionary equations is feasible, including Benjamin-Bona-Mahony-Burgers' equation, Korteweg-de Vries equation, Kuramoto-Sivashinsky equation, and classification to name a few. Numerical results demonstrate that the presented schemes for Burgers' equation can achieve second-order accuracy in time and fourth-order accuracy in space in \(L^{\infty}\)-norm.Uniform \(l^1\) behavior of the first-order interpolant quadrature scheme for some partial integro-differential equationshttps://zbmath.org/1472.651042021-11-25T18:46:10.358925Z"Xu, Da"https://zbmath.org/authors/?q=ai:xu.daIn this paper, the time discrete scheme based on the first-order backward difference method for the following integro-differential problem is studied \[ u_t(t)+\int\limits_0^t\beta(t-s)Au(s)ds=0, \quad t>0, \quad u(0)=u_0. \] Here \(A\) is a positive self-adjoint linear operator defined on a dense subspace \(D(A)\) of the real Hilbert space \(H\), \(u_0\in H\), and \(\beta(t)\) is weakly singular kernel at \(t=0\) such that \(\beta(t)=t^{\alpha-1}/\Gamma(\alpha)\), \(0<\alpha<1\). The paper is organized as follows. Section 1 is an introduction. In this section abovementioned problem is stated and the main theorem of this work is formulated. A review of the suitable papers is also given in this section. The memory term is approximated by the interpolating quadrature in this paper. In Section 2, using the Laplace transform technique is shown that this interpolating quadrature scheme has a convergence rate of \(O(\Delta t)\), where \(\Delta t\) denotes the time step. Special attention is that the uniform \(\ell^1\) convergence property of the discretization in time is given.Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methodshttps://zbmath.org/1472.651052021-11-25T18:46:10.358925Z"Bertaglia, Giulia"https://zbmath.org/authors/?q=ai:bertaglia.giulia"Pareschi, Lorenzo"https://zbmath.org/authors/?q=ai:pareschi.lorenzoIn this work, a novel SIR-type kinetic transport model for the spread of infectious diseases on networks is developed, where S is the number of susceptible individuals, I is the number of infected individuals, and R is the number of recovered individuals. The hyperbolic system describes at a macroscopic level the propagation of epidemics at finite speeds, recovering the classical one-dimensional reaction-diffusion model as relaxation times and characteristic speeds of each compartment of the population (susceptible, infectious and recovered individuals) tend to zero and infinity, respectively. As discretization for this interesting model, the authors suggest a finite-volume IMEX method able to maintain the consistency with the diffusive limit without restrictions due to the scaling parameters. Several numerical tests for simple epidemic network structures are presented to confirm the ability of the model to correctly describe the spread of an epidemic.Convergence analysis of the discrete duality finite volume scheme for the regularised Heston modelhttps://zbmath.org/1472.651062021-11-25T18:46:10.358925Z"Tibenský, Matúš"https://zbmath.org/authors/?q=ai:tibensky.matus"Handlovičová, Angela"https://zbmath.org/authors/?q=ai:handlovicova.angelaAuthors' abstract: The aim of the paper is to study the problem of financial derivatives pricing based on the idea of the Heston model introduced in [\textit{S. L. Heston}, Rev. Financ. Stud. 6, No. 2, 327--343 (1993; Zbl 1384.35131)]. The authors construct a regularised version of the Heston model and the discrete duality finite volume (DDFV) scheme for this model. The numerical analysis is performed for this scheme and stability estimates on the discrete solution and the discrete gradient are obtained. In addition, the convergence of the DDFV scheme to the weak solution of the regularised Heston model is proven. Numerical experiments are provided in the end of the paper to test the regularisation parameter impact.Kernel based high order ``explicit'' unconditionally stable scheme for nonlinear degenerate advection-diffusion equationshttps://zbmath.org/1472.651072021-11-25T18:46:10.358925Z"Christlieb, Andrew"https://zbmath.org/authors/?q=ai:christlieb.andrew-j"Guo, Wei"https://zbmath.org/authors/?q=ai:guo.wei"Jiang, Yan"https://zbmath.org/authors/?q=ai:jiang.yan"Yang, Hyoseon"https://zbmath.org/authors/?q=ai:yang.hyoseonIn the present article, authors propose a high-order numerical scheme for solving a class of nonlinear degenerate parabolic equations written as
\[
\partial_t u+\partial_x f(u)=\partial_{xx}g(u),
\]
where \(g'(u)\geq 0\) and \(g'(u)\) can vanish for some values of \(u\). This type of equations arises in a wide range of applications, containing for example radiative transport or porous medium flow. It has similar properties as hyperbolic conservation laws, including in particular possible existence of nonsmooth solutions. It is therefore important to consider a high order approximation for this type of equations. Moreover, since the problem is parabolic, a classical explicit time discretization would imply a constraining restriction on the time step to ensure stability, while classical implicit methods would lead to invert some operator at each time steps.
To overcome these issues, an approach based on the method of lines transpose (\(MOL^T\)) is considered. The idea is to first discretize the time, here with an explicit strong-stability-preserving Runge-Kutta (SSP RK) method, and then to solve a boundary value problem at each discrete time levels. Let us mention that the same ideas where used in [\textit{A. Christlieb} et al., J. Comput. Phys. 327, 337--367 (2016; Zbl 1422.65432)] for advection equations, the main difference being that in this previous article, an implicit SSP RK method was used instead of an explicit one. Then, the spatial derivatives are represented as infinite series, in which each term relies on a special kernel based formulation of the solution. This approach makes the method effectively implicit at each stage of the explicit SSP RK method, but without the need of invert any operator. Finally, a WENO methodology is applied, but may not be sufficient to suppress solution overshoots in the case of nonsmooth problems. To enhance robustness of the method, a nonlinear filter is introduced, constructed thanks to the smoothness indicators provided by the WENO methodology, and then not increasing the computational cost.
Unconditional stability of the proposed numerical scheme is rigorously proved in the linear case with periodic boundary conditions, for order \(k=1,\,2,\,3\). The stability for the more general nonlinear case is not theoretically established, but numerically confirmed. Several numerical 1D and 2D test cases, including porous medium equation, Buckley-Leverett equation and strongly degenerate parabolic problems, are presented. Numerical results demonstrate efficiency of the proposed method, high order accuracy and ability to produce nonoscillatory shock transitions. Moreover, it appears that the CFL number can be chosen arbitrarily large, leading only to a loss of accuracy and not a stability issue.Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equationshttps://zbmath.org/1472.651082021-11-25T18:46:10.358925Z"Wang, Wansheng"https://zbmath.org/authors/?q=ai:wang.wansheng"Mao, Mengli"https://zbmath.org/authors/?q=ai:mao.mengli"Wang, Zheng"https://zbmath.org/authors/?q=ai:wang.zhengSummary: In this paper, stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the \(l^{\infty}(0,T;H)\)-stability of the BDF2 method for linear and semilinear parabolic equations is identical with the upper bound for the zero-stability. The \(l^{\infty}(0,T;V)\)-stability of the variable step-size BDF2 method is also established under more relaxed condition on the ratios of consecutive step-sizes. Based on these stability results, error estimates in several different norms are derived. To utilize the BDF method, the trapezoidal method and the backward Euler scheme are employed to compute the starting value. For the latter choice, order reduction phenomenon of the constant step-size BDF2 method is observed theoretically and numerically in several norms. Numerical results also illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations.The hessian Riemannian flow and Newton's method for effective Hamiltonians and Mather measureshttps://zbmath.org/1472.651102021-11-25T18:46:10.358925Z"Gomes, Diogo A."https://zbmath.org/authors/?q=ai:gomes.diogo-luis-aguiar"Yang, Xianjin"https://zbmath.org/authors/?q=ai:yang.xianjinThe authors suggest two methods, the Hessian Riemannian flow and Newton's method, to calculate simultaneously the effective Hamiltonian and the Mather measures. The convergence of the Hessian Riemannian flow in the continuous setting is proved. For the discrete case, the existence and the convergence of the Hessian Riemannian flow are shown. A variant of Newton's method, which improves the performance of the Hessian Riemannian flow, is used. Some numerical tests are presented to show that the algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular.A local meshless method for the numerical solution of space-dependent inverse heat problemshttps://zbmath.org/1472.651112021-11-25T18:46:10.358925Z"Khan, Muhammad Nawaz"https://zbmath.org/authors/?q=ai:khan.muhammad-nawaz"Siraj-ul-Islam"https://zbmath.org/authors/?q=ai:siraj-ul-islam."Hussain, Iltaf"https://zbmath.org/authors/?q=ai:hussain.iltaf"Ahmad, Imtiaz"https://zbmath.org/authors/?q=ai:ahmad.imtiaz"Ahmad, Hijaz"https://zbmath.org/authors/?q=ai:ahmad.hijazThe authors consider an inverse problem for the space-wise dependent heat source problem. The source term, which is only depending on the spatial variable, and the unknown solution are to be computed while an overspecified condition is given when \(t=T\). A local radial basis function collocation method is proposed. Multiquadric radial basis functions are used as approximation for the spatial discretization. The method accuracy is tested in terms of absolute root mean square and relative root mean square error norms. Numerical tests on a noisy data are performed on both regular domain and irregular domain. A comparision with some existing methods is also made.Travelling wave solutions of multisymplectic discretizations of semi-linear wave equationshttps://zbmath.org/1472.651122021-11-25T18:46:10.358925Z"McDonald, Fleur"https://zbmath.org/authors/?q=ai:mcdonald.fleur"McLachlan, Robert I."https://zbmath.org/authors/?q=ai:mclachlan.robert-i"Moore, Brian E."https://zbmath.org/authors/?q=ai:moore.brian-e"Quispel, G. R. W."https://zbmath.org/authors/?q=ai:quispel.gilles-reinout-willemThe travelling waves are solutions to the PDEs (Partial Differential Equations) that travel at a constant speed \(c\) without changing shape. To understand preservation of multisymplectic discretisations of travelling wave solutions, the authors apply the 5-point central difference scheme to the semi-linear wave equation. A travelling wave ansatz leads to an ordinary difference equation whose solutions can be compared to travelling wave solutions of the PDE. Two parts should be handled. The first part is a discontinuous nonlinearity in which the difference equation is solved exactly. For continuous nonlinearities, the difference equation is solved using a Fourier series, and resonances that depend on the grid-size are revealed for a smooth nonlinearity. In general, the infinite dimensional functional equation, which must be solved to get the travelling wave solutions, is intractable, but backward error analysis proves to be a powerful tool, as it provides a way to study the solutions of equation through a simple ordinary differential equation that describes the behavior to arbitrarily high order. A general framework for using backward error analysis to analyze preservation of travelling waves for other equations and discretisations is presented.Analytical and Rothe time-discretization method for a Boussinesq-type system over an uneven bottomhttps://zbmath.org/1472.651132021-11-25T18:46:10.358925Z"Mejía, Luis Fernando"https://zbmath.org/authors/?q=ai:mejia.luis-fernando"Muñoz Grajales, Juan Carlos"https://zbmath.org/authors/?q=ai:munoz-grajales.juan-carlosThe authors consider the analytical and numerical resolution of a 2D version of a Boussinesq-type model which occur in the water wave propagation. The time discretization is performed using a finite-difference second-order Crank-Nicholson-type scheme, and then, at each time step, the spatial variables are discretized with an efficient Galerkin/Finite Element Method (FEM) using triangular-finite elements based on 2D piecewise-linear Lagrange interpolation. Some numerical tests are presented to support the theoretical results.Active force generation in cardiac muscle cells: mathematical modeling and numerical simulation of the actin-myosin interactionhttps://zbmath.org/1472.651142021-11-25T18:46:10.358925Z"Regazzoni, Francesco"https://zbmath.org/authors/?q=ai:regazzoni.francesco.1|regazzoni.francesco"Dedè, Luca"https://zbmath.org/authors/?q=ai:dede.luca"Quarteroni, Alfio"https://zbmath.org/authors/?q=ai:quarteroni.alfio-mThe authors first review several models describing the interaction between actin and myosin in cardiac muscle cells. The most detailed models are able of capturing phenomena, such as the response to fast steps, occurring at the fastest time scales involved in the force generation mechanism. The issue of parameter identifiability for force generation models is considered. In particular, the authors show, for a modified version of the Huxley model, how the model parameters can be estimated by measurements typically available from experiments.New modification of Adomian decomposition method for solving a system of nonlinear fractional partial differential equationshttps://zbmath.org/1472.651152021-11-25T18:46:10.358925Z"Thabet, Hayman"https://zbmath.org/authors/?q=ai:thabet.hayman"Kendre, Subhash"https://zbmath.org/authors/?q=ai:kendre.s-d|kendre.subhash-dhondibaIn this paper, a new modification of ADM (Adomian Decomposition Method) for solving a fully general system of NFPDEs is introduced. The convergence and error analysis of the proposed modification are presented. In addition to this, it is shown that the approximate analytical and numerical solutions are in very good conformity with the exact solutions that previously obtained by some other well-known methods. Some numerical results are carried out to confirm the accuracy and efficiency of this new ADM.Application of the cut-off projection to solve a backward heat conduction problem in a two-slab composite systemhttps://zbmath.org/1472.651162021-11-25T18:46:10.358925Z"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Khoa, Vo Anh"https://zbmath.org/authors/?q=ai:khoa.vo-anh"Truong, Mai Thanh Nhat"https://zbmath.org/authors/?q=ai:truong.mai-thanh-nhat"Hung, Tran The"https://zbmath.org/authors/?q=ai:hung.tran-the"Minh, Mach Nguyet"https://zbmath.org/authors/?q=ai:mach-nguyet-minh.Summary: The main goal of this paper is applying the cut-off projection for solving one-dimensional backward heat conduction problem in a two-slab system with a perfect contact. In a constructive manner, we commence by demonstrating the Fourier-based solution that contains the drastic growth due to the high-frequency nature of the Fourier series. Such instability leads to the need of studying the projection method where the cut-off approach is derived consistently. In the theoretical framework, the first two objectives are to construct the regularized problem and prove its stability for each noise level. Our second interest is estimating the error in \(L^2\)-norm. Another supplementary objective is computing the eigen-elements. All in all, this paper can be considered as a preliminary attempt to solve the heating/cooling of a two-slab composite system backward in time. Several numerical tests are provided to corroborate the qualitative analysis.New approach to identify the initial condition in degenerate hyperbolic equationhttps://zbmath.org/1472.651172021-11-25T18:46:10.358925Z"Atifi, K."https://zbmath.org/authors/?q=ai:atifi.khalid"Essoufi, El-H."https://zbmath.org/authors/?q=ai:essoufi.el-hassan"Khouiti, B."https://zbmath.org/authors/?q=ai:khouiti.bouchraSummary: This paper deals with the determination of an initial condition in degenerate hyperbolic equation from final observations. With the aim of reducing the execution time, this inverse problem is solved using an approach based on double regularization: a Tikhonov's regularization and regularization in equation by viscose-elasticity. So, we obtain a sequence of weak solutions of degenerate linear viscose-elastic problems. Firstly, we prove the existence and uniqueness of each term of this sequence. Secondly, we prove the convergence of this sequence to the weak solution of the initial problem. Also we present some numerical experiments to show the performance of this approach.Collocation method based on shifted Chebyshev and radial basis functions with symmetric variable shape parameter for solving the parabolic inverse problemhttps://zbmath.org/1472.651192021-11-25T18:46:10.358925Z"Ranjbar, Mojtaba"https://zbmath.org/authors/?q=ai:ranjbar.mojtaba"Aghazadeh, Mansour"https://zbmath.org/authors/?q=ai:aghazadeh.mansourSummary: This work introduces a new numerical solution to the inverse parabolic problem with source control parameter that has important applications in large fields of applied science. We expand the approximate solution of the inverse problem in terms of shifted Chebyshev polynomials in time and radial basis functions with symmetric variable shape parameter in space, with unknown coefficients. Unknown coefficient matrix determined using the collocation technique. Sample results show that the proposed method is very accurate. Moreover, the proposed method is compared with two other methods, fourth-order compact difference scheme and method of lines. Finally, we examine the stability of our method for the case where there is additive noise in input data.Optimal a priori error estimates for the finite element approximation of dual-phase-lag bio heat model in heterogeneous mediumhttps://zbmath.org/1472.651202021-11-25T18:46:10.358925Z"Dutta, Jogen"https://zbmath.org/authors/?q=ai:dutta.jogen"Deka, Bhupen"https://zbmath.org/authors/?q=ai:deka.bhupenSummary: Galerkin finite element method is applied to dual-phase-lag bio heat model in heterogeneous medium. Well-posedness of the model interface problem and a priori estimates of its solutions are established. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in \(L^\infty (L^2)\) norm. The fully discrete space-time finite element discretizations is based on second order in time Newmark scheme. Finally, numerical results for two dimensional test problems are presented in support of our theoretical findings. Finite element algorithm presented here can contribute to a variety of engineering and medical applications.Non-intrusive reduced-order modeling of parameterized electromagnetic scattering problems using cubic spline interpolationhttps://zbmath.org/1472.651222021-11-25T18:46:10.358925Z"Li, Kun"https://zbmath.org/authors/?q=ai:li.kun.5"Huang, Ting-Zhu"https://zbmath.org/authors/?q=ai:huang.ting-zhu"Li, Liang"https://zbmath.org/authors/?q=ai:li.liang"Lanteri, Stéphane"https://zbmath.org/authors/?q=ai:lanteri.stephaneSummary: This paper presents a non-intrusive model order reduction (MOR) for the solution of parameterized electromagnetic scattering problems, which needs to prepare a database offline of full-order solution samples (snapshots) at some different parameter locations. The snapshot vectors are produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. Because the second dimension of snapshots matrix is large, a two-step or nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the projection coefficient matrices (also referred to as the reduced coefficient matrices) of full-order solutions onto the RB subspace are extracted. A cubic spline interpolation-based (CSI) approach is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices without resorting to Galerkin projection. The generation of snapshot vectors, the construction of POD basis functions and the approximation of reduced coefficient matrices based on the CSI method are completed during the offline stage. The RB solutions for new time and parameter values can be rapidly recovered via outputs from the interpolation models in the online stage. In particular, the offline and online stages of the proposed RB method, termed as the POD-CSI method, are completely decoupled, which ensures the computational validity of the method. Moreover, a surrogate error model is constructed as an efficient error estimator for the POD-CSI method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of POD-CSI method.Fractional Crank-Nicolson-Galerkin finite element methods for nonlinear time fractional parabolic problems with time delayhttps://zbmath.org/1472.651232021-11-25T18:46:10.358925Z"Li, Lili"https://zbmath.org/authors/?q=ai:li.lili"She, Mianfu"https://zbmath.org/authors/?q=ai:she.mianfu"Niu, Yuanling"https://zbmath.org/authors/?q=ai:niu.yuanlingSummary: A linearized numerical scheme is proposed to solve the nonlinear time-fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson method, and extrapolation methods in the temporal direction. A novel discrete fractional Grönwall inequality is established. Thanks to the inequality, the error estimate of a fully discrete scheme is obtained. Several numerical examples are provided to verify the effectiveness of the fully discrete numerical method.A conservative finite element ALE scheme for mass-conservative reaction-diffusion equations on evolving two-dimensional domainshttps://zbmath.org/1472.651242021-11-25T18:46:10.358925Z"Mackenzie, John"https://zbmath.org/authors/?q=ai:mackenzie.john-a"Rowlatt, Christopher"https://zbmath.org/authors/?q=ai:rowlatt.christopher"Insall, Robert"https://zbmath.org/authors/?q=ai:insall.robertThe paper is concerned with the numerical solution of mass-conservative bulk-surface reaction-diffusion equations on evolving two-dimensional domains. These types of equations have recently been shown to represent intracellular pattern formation. The authors propose a method that is a combination of a conservative arbitrary Lagrangian-Eulerian finite element method for spatial discretization and a second-order two-stage time integration scheme for temporal discretization. It is proved that the fully discrete scheme is globally conservative independently of the Lagrangian-Eulerian velocity and time step size. Numerical experiments are presented to illustrate the efficiency of the scheme and to confirm the second-order convergence and global conservation. Finally, the proposed method is applied to the so-called WP model for directed cell migration.High-order BDF fully discrete scheme for backward fractional Feynman-Kac equation with nonsmooth datahttps://zbmath.org/1472.651262021-11-25T18:46:10.358925Z"Sun, Jing"https://zbmath.org/authors/?q=ai:sun.jing"Nie, Daxin"https://zbmath.org/authors/?q=ai:nie.daxin"Deng, Weihua"https://zbmath.org/authors/?q=ai:deng.weihuaSummary: The Feynman-Kac equation governs the distribution of the statistical observable -- functional, having wide applications in almost all disciplines. After overcoming some challenges from the time-space coupled nonlocal operator and the possible low regularity of functional, this paper develops the high-order fully discrete scheme for the backward fractional Feynman-Kac equation by using backward difference formulas (BDF) convolution quadrature in time, finite element method in space, and some correction terms. With a systematic correction, the high convergence order is achieved up to 6 in time, without deteriorating the optimal convergence in space and without the regularity requirement on the solution. Finally, the extensive numerical experiments validate the effectiveness of the high-order schemes.High precision numerical approach for Davey-Stewartson II type equations for Schwartz class initial datahttps://zbmath.org/1472.651272021-11-25T18:46:10.358925Z"Klein, Christian"https://zbmath.org/authors/?q=ai:klein.christian"McLaughlin, Ken"https://zbmath.org/authors/?q=ai:mclaughlin.kenneth-d-t-r"Stoilov, Nikola"https://zbmath.org/authors/?q=ai:stoilov.nikola-mSummary: We present an efficient high-precision numerical approach for Davey-Stewartson (DS) II type equa- tions, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll's composite Runge-Kutta method for the time dependence. Since DS equations are non-local, nonlinear Schrödinger equations with a singular symbol for the non-locality, standard Fourier methods in practice only reach accuracy of the order of \(10^{-6}\) or less for typical examples. This was previously demonstrated for the defocusing integrable case by comparison with a numerical approach for DS II via inverse scattering. By applying a regularization to the singular symbol, originally developed for D-bar problems, the presented code is shown to reach machine precision. The code can treat integrable and non-integrable DS II equations. Moreover, it has the same numerical complexity as existing codes for DS II. Several examples for the integrable defocusing DS II equation are discussed as test cases. In an appendix by C. Kalla, a doubly periodic solution to the defocusing DS II equation is presented, providing a test for direct DS codes based on Fourier methods.Numerical-solution-for-nonlinear-Klein-Gordon equation via operational-matrix by clique polynomial of complete graphshttps://zbmath.org/1472.651282021-11-25T18:46:10.358925Z"Kumbinarasaiah, S."https://zbmath.org/authors/?q=ai:kumbinarasaiah.s"Ramane, H. S."https://zbmath.org/authors/?q=ai:ramane.harishchandra-s"Pise, K. S."https://zbmath.org/authors/?q=ai:pise.kartik-s"Hariharan, G."https://zbmath.org/authors/?q=ai:hariharan.govindSummary: This study introduced a generalized operational matrix using Clique polynomials of a complete graph and proposed the latest approach to solve the non-linear Klein-Gordon (KG) equation. KG equations describe many real physical phenomena in fluid dynamics, electrical engineering, biogenetics, tribology. By using the properties of the operational-matrix, we transform-the non-linear KG equation into a system-of algebraic-equations. Unknown coefficients to be determined by Newton's method. The present-technique is applied-to four problems, and the obtained-results are-compared with-another-method in the literature. Also, we discussed some theorems on convergence analysis and continuous property.An adaptive spectral graph wavelet method for PDEs on networkshttps://zbmath.org/1472.651292021-11-25T18:46:10.358925Z"Mehra, Mani"https://zbmath.org/authors/?q=ai:mehra.mani"Shukla, Ankita"https://zbmath.org/authors/?q=ai:shukla.ankita"Leugering, Günter"https://zbmath.org/authors/?q=ai:leugering.gunterSummary: In this article, we propose an adaptive spectral graph wavelet method to solve partial differential equations on network-like structures using so-called spectral graph wavelets. The concept of spectral graph wavelets is based on the discrete graph Laplacian. The beauty of the method lies in the fact that the same operator is used for the approximation of differential operators and for the construction of the spectral graph wavelets. Two test functions on different topologies of the network are considered in order to explain the features of the spectral graph wavelet (i.e., behavior of wavelet coefficients and reconstruction error). Subsequently, the method is applied to parabolic problems on networks with different topologies. The numerical results show that the method accurately captures the emergence of the localized patterns at all the scales (including the junction of the network) and the node arrangement is accordingly adapted. The convergence of the method is verified and the efficiency of the method is discussed in terms of CPU time.A new numerical approach for single rational soliton solution of Chen-Lee-Liu equation with Riesz fractional derivative in optical fibershttps://zbmath.org/1472.651302021-11-25T18:46:10.358925Z"Ray, Santanu Saha"https://zbmath.org/authors/?q=ai:saha-ray.santanuThe author proposes a time-splitting spectral approximation technique for the Chen-Lee-Liu (CLL) equation involving Riesz fractional derivative, which is the nonlinear Schrodinger equation with Riesz fractional derivative, based on the Strang splitting spectral method. The proposed numerical technique is shown to be efficient, unconditionally stable, and of second-order accuracy in time and of spectral accuracy in space. It conserves the total density in the discretized level. To examine the results, the author also gives a Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme for the Riesz fractional CLL equation. By the comparison of numerical results, the proposed time-splitting spectral method shows to be effective and simple for obtaining single soliton numerical solution of the Riesz fractional CLL equation.Efficient computational approach for generalized fractional KdV-Burgers equationhttps://zbmath.org/1472.651312021-11-25T18:46:10.358925Z"Rida, Saad Z."https://zbmath.org/authors/?q=ai:rida.saad-zagloul"Hussien, Hussien S."https://zbmath.org/authors/?q=ai:hussien.hussien-shafeiSummary: A collocation method based on double summations of Mittag-Leffler functions is proposed to solve the Korteweg-de Vries (KdV) and Burgers equation of fractional order with initial-boundary conditions. The resulting algebraic system is constructed as a constrained optimization problem and optimized to obtain the unknown coefficients. Error analysis of the approximation solution is studied. Simulations of the results are studied graphically through representations for the effect of fractional order parameters and time levels. The results ensure that the proposed method is accurate and efficient.Numerical solution of nonlinear fifth-order KdV-type partial differential equations via Haar wavelethttps://zbmath.org/1472.651322021-11-25T18:46:10.358925Z"Saleem, Sidra"https://zbmath.org/authors/?q=ai:saleem.sidra"Hussain, Malik Zawwar"https://zbmath.org/authors/?q=ai:hussain.malik-zawwarSummary: In this research article, the numerical solution of different forms of widely used one-dimensional Korteweg-de Vries equation is discussed. For this purpose, Haar wavelet collocation method is implemented. A simple algorithm is constructed which is based on the proposed method. The presented method is tested on fifth-order Lax equation, Sawada-Kotera equation, Caudrey-Dodd-Gibbon equation, Kaup-Kuperschmidt equation and Ito equation. The obtained approximate results are displayed using tables and figures. The numerical results show good accuracy of the proposed method.Computational scheme for the time-fractional reaction-diffusion Brusselator modelhttps://zbmath.org/1472.651332021-11-25T18:46:10.358925Z"Alquran, Marwan"https://zbmath.org/authors/?q=ai:alquran.marwan-taiseer"Jaradat, Imad"https://zbmath.org/authors/?q=ai:jaradat.imad"Ali, Mohammed"https://zbmath.org/authors/?q=ai:ali.mohammed-ashraf|ali.mohammed-l|ali.mohammed-shabbir|ali.mohammed-hasan|ali.mohammed-eunus|ali.mohammed-yousef-ahmad"Abu Aljazar, Ahlam"https://zbmath.org/authors/?q=ai:abu-aljazar.ahlamSummary: In this work, we present an adaptation of a new look of the fractional Maclaurin series to study the time-fractional reaction-diffusion Brusselator system. Also, we give a description of implementing the suggested numerical scheme to provide a supportive approximation solution to the time-fractional Brusselator. We study the physical shape of the depicted solution upon changing the order of the fractional derivative and concluding some results. The analysis conducted in this work is supported by 2D-3D plots. Finally, we discuss other numerical techniques that has been used in obtaining simulated solutions for the fractional Brusselator model.Efficient approximation of solutions of parametric linear transport equations by ReLU DNNshttps://zbmath.org/1472.651342021-11-25T18:46:10.358925Z"Laakmann, Fabian"https://zbmath.org/authors/?q=ai:laakmann.fabian"Petersen, Philipp"https://zbmath.org/authors/?q=ai:petersen.philipp-cSummary: We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension.The mollification method based on a modified operator to the ill-posed problem for 3D Helmholtz equation with mixed boundaryhttps://zbmath.org/1472.651352021-11-25T18:46:10.358925Z"He, Shangqin"https://zbmath.org/authors/?q=ai:he.shangqin"Di, Congna"https://zbmath.org/authors/?q=ai:di.congna"Yang, Li"https://zbmath.org/authors/?q=ai:yang.li.3|yang.li.2|yang.li|yang.li.1In this work, a Cauchy problem for the 3D Helmholtz equation with mixed boundary is studied. The mollification method based on modified bivariate de la Vallée Poussin operator to solve the stated Cauchy problem is used. It is verified that the considered method is stable. The paper is organized as follows. Section 1 is an introduction. In Section 2, the ill-posedness of the considered problem is illustrated. The modified bivariate de la Vallée Poussin kernel and its certain properties are presented in Section 3. This is also used to construct the mollification operator to obtain the regularization solution. Section 4 is devoted to the stable convergence estimates under the suitable choices of regularization parameters in \(0<z<d\) and at boundary \(z=d\). The result of two numerical experiments are presented with tables and graphical illustrations in Section 5. The efficiency of the method is demonstrated too. Some conclusions are given in Section 6.On the use of the adjoint operator for source reconstruction in particle transport problemshttps://zbmath.org/1472.651362021-11-25T18:46:10.358925Z"Pazinatto, C. B."https://zbmath.org/authors/?q=ai:pazinatto.c-b"Barichello, L. B."https://zbmath.org/authors/?q=ai:barichello.liliane-bassoSummary: In this work, the adjoint to the transport operator is used to estimate the spatial distribution of an isotropic neutral particles source in a homogeneous one-dimensional medium, from readings of internal detectors. An analytical discrete ordinates formulation, the ADO method, is applied to derive a spatially explicit solution for the adjoint flux. Simulations are performed for reconstructing Gaussian and piecewise localized sources. Numerical results indicate that the source estimates may be considered satisfactory although scalar fluxes are reconstructed more precisely.Non-conforming Crouzeix-Raviart element approximation for Stekloff eigenvalues in inverse scatteringhttps://zbmath.org/1472.651372021-11-25T18:46:10.358925Z"Yang, Yidu"https://zbmath.org/authors/?q=ai:yang.yidu"Zhang, Yu"https://zbmath.org/authors/?q=ai:zhang.yu.3"Bi, Hai"https://zbmath.org/authors/?q=ai:bi.haiSummary: In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-self-adjoint and indefinite, and its Crouzeix-Raviart element discretization does not meet the condition of the Strang lemma. We use the standard duality technique to prove an extension of the Strang lemma. And we prove the convergence and error estimate of discrete eigenvalues and eigenfunctions using the spectral perturbation theory for compact operators. Finally, we present some numerical examples not only on uniform meshes but also on adaptive refined meshes to show that the Crouzeix-Raviart method is efficient for computing real and complex eigenvalues as expected.Analysis of the shifted boundary method for the Poisson problem in domains with cornershttps://zbmath.org/1472.651402021-11-25T18:46:10.358925Z"Atallah, Nabil M."https://zbmath.org/authors/?q=ai:atallah.nabil-m"Canuto, Claudio"https://zbmath.org/authors/?q=ai:canuto.claudio"Scovazzi, Guglielmo"https://zbmath.org/authors/?q=ai:scovazzi.guglielmoThe authors are concerned with the shifted boundary method (SBM) in order to accurately solve the Poisson problem with non homogenous Dirichlet boundary conditions in rather general domains in two or three dimensions, i.e., domains exhibiting corners and edges. First, the authors introduce the method and precisely state how the exact boundary condition is mapped into the surrogate boundary. Then they establish the coercivity and continuity properties of the SBM variational formulation and analyze the behavior of the Taylor remainder on the surrogate boundary. Using the Strang's Second Lemma the authors obtain consistency and convergence results (in the energy norm) and eventually they transfer these results in $L^2$ norm via a duality argument. A very suggestive example is carried out.\(L^\infty\) Norm error estimates for HDG methods applied to the Poisson equation with an application to the Dirichlet boundary control problemhttps://zbmath.org/1472.651422021-11-25T18:46:10.358925Z"Chen, Gang"https://zbmath.org/authors/?q=ai:chen.gang.1"Monk, Peter B."https://zbmath.org/authors/?q=ai:monk.peter-b"Zhang, Yangwen"https://zbmath.org/authors/?q=ai:zhang.yangwenIn this paper quasi-optimal \(L^\infty\) norm error estimates for the standard hybridizable discontinuous Galerkin (HDG) method for the solution of Poisson's problem in two-dimensional space is proved. The paper is organized as follows. Section 1 is an introduction. In Section 2, HDG formulation and preliminary material is given. The main result of this section is to extend the \(L^2\) norm estimates for the auxiliary projections used in the error analysis of HDG to \(L^p\) norms \((1\le p \le\infty\). The main result of this paper, \(L^\infty\) norm estimates are proved in Section 3. Quasi-optimal estimates on interfaces are given in Section 4. An elliptic Dirichlet boundary control problem is considered in Section 5. Two numerical examples with tables to illustrate theoretical results are presented in Section 6. Some conclusions are given in Section 7.Isoparametric finite element analysis of a generalized Robin boundary value problem on curved domainshttps://zbmath.org/1472.651432021-11-25T18:46:10.358925Z"Edelmann, Dominik"https://zbmath.org/authors/?q=ai:edelmann.dominikSummary: We study the discretization of an elliptic partial differential equation, posed on a two- or three-dimensional domain with smooth boundary, endowed with a generalized Robin boundary condition which involves the Laplace-Beltrami operator on the boundary surface. The boundary is approximated with piecewise polynomial faces and we use isoparametric finite elements of arbitrary order for the discretization. We derive optimal-order error bounds for this non-conforming finite element method in both \(L^2\)- and \(H^1\)-norm. Numerical examples illustrate the theoretical results.Optimal quadratic element on rectangular grids for \(H^1\) problemshttps://zbmath.org/1472.651502021-11-25T18:46:10.358925Z"Zeng, Huilan"https://zbmath.org/authors/?q=ai:zeng.huilan"Zhang, Chen-Song"https://zbmath.org/authors/?q=ai:zhang.chensong"Zhang, Shuo"https://zbmath.org/authors/?q=ai:zhang.shuoSummary: In this paper, a piecewise quadratic finite element method on rectangular grids for \(H^1\) problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is proved to be \(O(h^2)\) in the energy norm on uniform grids over a convex domain. A lower bound of the \(L^2\)-norm error is also proved, which makes the capacity of this scheme more clear. For the eigenvalue problem, the computed eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented to verify the theoretical findings.Decoupling PDE computation with intrinsic or inertial Robin interface conditionhttps://zbmath.org/1472.651512021-11-25T18:46:10.358925Z"Zhang, Lian"https://zbmath.org/authors/?q=ai:zhang.lian"Cai, Mingchao"https://zbmath.org/authors/?q=ai:cai.mingchao"Mu, Mo"https://zbmath.org/authors/?q=ai:mu.moThe authors study decoupled numerical methods for a typical multi-modeling model problem. By cruising various stages of numerical approximation and decoupling, and tracking how the information is transmitted across the interface, the authors demonstrate how the interface conditions evolve and affect the data and error transmission across the interface during spatial/temporal discretization as well as decoupling. Moreover, the authors derive a new approximate intrinsic or inertial type Robin condition for a semi-discrete model with finite element discretization, which is justified both mathematically and physically in contrast to the classical Robin interface condition conventionally introduced for decoupling multi-modeling problems. Based on this new interface condition, an equivalent semi-discrete model is introduced, which provides a general framework for devising effective decoupled numerical methods. Numerical experiments also confirm the effectiveness and robustness of the decoupling approach proposed in this paper.Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element methodhttps://zbmath.org/1472.651522021-11-25T18:46:10.358925Z"Pintore, Moreno"https://zbmath.org/authors/?q=ai:pintore.moreno"Pichi, Federico"https://zbmath.org/authors/?q=ai:pichi.federico"Hess, Martin"https://zbmath.org/authors/?q=ai:hess.martin-wilfried"Rozza, Gianluigi"https://zbmath.org/authors/?q=ai:rozza.gianluigi"Canuto, Claudio"https://zbmath.org/authors/?q=ai:canuto.claudioSummary: The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.Erratum to: ``A survey of Trefftz methods for the Helmholtz equation''https://zbmath.org/1472.651542021-11-25T18:46:10.358925Z"Hiptmair, Ralf"https://zbmath.org/authors/?q=ai:hiptmair.ralf"Moiola, Andrea"https://zbmath.org/authors/?q=ai:moiola.andrea"Perugia, Ilaria"https://zbmath.org/authors/?q=ai:perugia.ilariaErratum to the authors' paper [ibid. 114, 237--279 (2016; Zbl 1357.65282)].
For the entire collection see [Zbl 1361.65002].A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equationshttps://zbmath.org/1472.651552021-11-25T18:46:10.358925Z"Farrell, Patrick E."https://zbmath.org/authors/?q=ai:farrell.patrick-e|farrell.patrick-emmet"Mitchell, Lawrence"https://zbmath.org/authors/?q=ai:mitchell.lawrence"Scott, L. Ridgway"https://zbmath.org/authors/?q=ai:scott.larkin-ridgway"Wechsung, Florian"https://zbmath.org/authors/?q=ai:wechsung.florianSummary: Augmented Lagrangian preconditioners have successfully yielded Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes equations, but only for specific discretizations. The discretizations for which these preconditioners have been designed possess error estimates which depend on the Reynolds number, with the discretization error deteriorating as the Reynolds number is increased. In this paper we present an augmented Lagrangian preconditioner for the Scott-Vogelius discretization on barycentrically-refined meshes. This achieves both Reynolds-robust performance and Reynolds-robust error estimates. A key consideration is the design of a suitable space decomposition that captures the kernel of the grad-div term added to control the Schur complement; the same barycentric refinement that guarantees inf-sup stability also provides a local decomposition of the kernel of the divergence. The robustness of the scheme is confirmed by numerical experiments in two and three dimensions.Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equationshttps://zbmath.org/1472.651572021-11-25T18:46:10.358925Z"Hutzenthaler, Martin"https://zbmath.org/authors/?q=ai:hutzenthaler.martin"Jentzen, Arnulf"https://zbmath.org/authors/?q=ai:jentzen.arnulf"Kruse, Thomas"https://zbmath.org/authors/?q=ai:kruse.thomas"Nguyen, Tuan Anh"https://zbmath.org/authors/?q=ai:nguyen.tuan-anh"von Wurstemberger, Philippe"https://zbmath.org/authors/?q=ai:von-wurstemberger.philippeSummary: For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy.Numerical solution of scattering problems using a Riemann-Hilbert formulationhttps://zbmath.org/1472.651762021-11-25T18:46:10.358925Z"Smith, Stefan G. Llewellyn"https://zbmath.org/authors/?q=ai:llewellyn-smith.stefan-g"Luca, Elena"https://zbmath.org/authors/?q=ai:luca.elenaSummary: A fast and accurate numerical method for the solution of scalar and matrix Wiener-Hopf (WH) problems is presented. The WH problems are formulated as Riemann-Hilbert problems on the real line, and a numerical approach developed for these problems is used. It is shown that the known far-field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results. A number of scalar and matrix WH problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the approach.Deep neural networks for waves assisted by the Wiener-Hopf methodhttps://zbmath.org/1472.681742021-11-25T18:46:10.358925Z"Huang, Xun"https://zbmath.org/authors/?q=ai:huang.xunSummary: In this work, the classical Wiener-Hopf method is incorporated into the emerging deep neural networks for the study of certain wave problems. The essential idea is to use the first-principle-based analytical method to efficiently produce a large volume of datasets that would supervise the learning of data-hungry deep neural networks, and to further explain the working mechanisms on underneath. To demonstrate such a combinational research strategy, a deep feed-forward network is first used to approximate the forward propagation model of a duct acoustic problem, which can find important aerospace applications in aeroengine noise tests. Next, a convolutional type U-net is developed to learn spatial derivatives in wave equations, which could help to promote computational paradigm in mathematical physics and engineering applications. A couple of extensions of the U-net architecture are proposed to further impose possible physical constraints. Finally, after giving the implementation details, the performance of the neural networks are studied by comparing with analytical solutions from the Wiener-Hopf method. Overall, the Wiener-Hopf method is used here from a totally new perspective and such a combinational research strategy shall represent the key achievement of this work.Erratum to: ``Constraint algorithm for singular field theories in the \(k\)-cosymplectic framework''https://zbmath.org/1472.700422021-11-25T18:46:10.358925Z"Gràcia, Xavier"https://zbmath.org/authors/?q=ai:gracia.xavier"Rivas, Xavier"https://zbmath.org/authors/?q=ai:rivas.xavier"Román-Roy, Narciso"https://zbmath.org/authors/?q=ai:roman-roy.narcisoFrom the text: A simple geometric description of singular autonomous field theories is provided
by \(k\)-presymplectic geometry. Consistency of field equations can be analyzed by
means of a constraint algorithm. In our recent paper [ibid. 12, No. 1, 1--23 (2020; Zbl 1434.70047)] we extended this analysis
to the non-autonomous case. In this case the geometric setting is provided by the
notion of \(k\)-precosymplectic structure. However, to ensure the existence of Reeb vector fields and Darboux coordinates, we restricted our attention to \(k\)-precosymplectic
manifolds of the form \(\mathbb{R}^k \times P\), with \(P\) a \(k\)-presymplectic manifold.
As a typical example, we analized the case of affine Lagrangians of the type
\[
L(x^\alpha,q^i,v^i_\alpha) = f^\mu_j(x^\alpha,q^i)v^j_\mu + g(x^\alpha,q^i)
\]
on the manifold \(\mathbb{R}^k \times T^1_k Q\), and a particular academic example (sections 6.1 and
6.2). Nevertheless, such Lagrangians do not result in \(k\)-precosymplectic structures
of the above mentioned type, and their analysis as presented in the paper is not
correct (for instance, Reeb vector fields may not be well defined).
In this note we correct this mistake by restricting our study to the family of
affine Lagrangians of the type \(L(x^\alpha,q^i,v^i_\alpha) = f(q^i)v^j_\mu + g(x^\alpha,q^i)\), which lead to
\(k\)-precosymplectic structures as previously indicated. We also analyze a particular
example in this class that replaces the one in section 6.2.Exponential stability in Mindlin's form II gradient thermoelasticity with microtemperatures of type IIIhttps://zbmath.org/1472.740062021-11-25T18:46:10.358925Z"Aouadi, M."https://zbmath.org/authors/?q=ai:aouadi.moncef"Passarella, F."https://zbmath.org/authors/?q=ai:passarella.francesca"Tibullo, V."https://zbmath.org/authors/?q=ai:tibullo.vincenzoSummary: In this paper, we derive a nonlinear strain gradient theory of thermoelastic materials with microtemperatures taking into account micro-inertia effects as well. The elastic behaviour is assumed to be consistent with Mindlin's Form II gradient elasticity theory, while the thermal behaviour is based on the entropy balance of type III postulated by Green and Naghdi for both temperature and microtemperatures. The work is motivated by increasing use of materials having microstructure at both mechanical and thermal levels. The equations of the linear theory are also obtained. Then, we use the semigroup theory to prove the well-posedness of the obtained problem. Because of the coupling between high-order derivatives and microtemperatures, the obtained equations do not have exponential decay. A frictional damping for the elastic component, whose form depends on the micro-inertia, is shown to lead to exponential stability for the type III model.Spatial estimates for Kelvin-Voigt finite elasticity with nonlinear viscosity: well behaved solutions in spacehttps://zbmath.org/1472.740402021-11-25T18:46:10.358925Z"Quintanilla, Ramon"https://zbmath.org/authors/?q=ai:quintanilla.ramon"Saccomandi, Giuseppe"https://zbmath.org/authors/?q=ai:saccomandi.giuseppeThe heat conduction problem for an isotropic plate with a heat-permeable cuthttps://zbmath.org/1472.740552021-11-25T18:46:10.358925Z"Bondarenko, N. S."https://zbmath.org/authors/?q=ai:bondarenko.n-sSummary: The problem of heat conduction for an isotropic plate with a heat-permeable cut based on the generalized theory is reduced to a system of independent boundary value problems for metaharmonic equations. The case of symmetric heat exchange is considered. The influence of parameters of heat exchange and heat permeability of the cut on the jumps of components for the perturbed temperature is investigated.Theory of the flow-induced deformation of shallow compliant microchannels with thick wallshttps://zbmath.org/1472.740652021-11-25T18:46:10.358925Z"Wang, Xiaojia"https://zbmath.org/authors/?q=ai:wang.xiaojia"Christov, Ivan C."https://zbmath.org/authors/?q=ai:christov.ivan-cSummary: Long, shallow microchannels embedded in thick, soft materials are widely used in microfluidic devices for lab-on-a-chip applications. However, the bulging effect caused by fluid-structure interactions between the internal viscous flow and the soft walls has not been completely understood. Previous models either contain a fitting parameter or are specialized to channels with plate-like walls. This work is a theoretical study of the steady-state response of a compliant microchannel with a thick wall. Using lubrication theory for low-Reynolds-number flows and the theory for linearly elastic isotropic solids, we obtain perturbative solutions for the flow and deformation. Specifically, only the channel's top wall deformation is considered, and the ratio between its thickness \(t\) and width \(w\) is assumed to be \((t/w)^2 \gg 1\). We show that the deformation at each stream-wise cross section can be considered independently, and that the top wall can be regarded as a simply supported rectangle subject to uniform pressure at its bottom. The stress and displacement fields are found using Fourier series, based on which the channel shape and the hydrodynamic resistance are calculated, yielding a new flow rate-pressure drop relation without fitting parameters. Our results agree favourably with, and thus rationalize, previous experiments.Mixed- and crack-type dynamical problems of electro-magneto-elasticity theoryhttps://zbmath.org/1472.740672021-11-25T18:46:10.358925Z"Buchukuri, Tengiz"https://zbmath.org/authors/?q=ai:buchukuri.tengiz"Chkadua, Otar"https://zbmath.org/authors/?q=ai:chkadua.otar"Natroshvili, David"https://zbmath.org/authors/?q=ai:natroshvili.davidThe authors investigate the solvability of three-dimensional dynamical mixed boundary value problems of electro-magneto-elasticity theory for homogeneous anisotropic bodies with interior cracks. Laplace transform technique, the potential method and the theory of pseudo-differential equations are used to prove the existence, uniqueness and regularity theorems for the considered problem, and to analyze asymptotic properties of solutions near the crack edges and near the intersection of lines carrying different boundary conditions.
There is a full coupling between the mechanical and electromagnetic fields. For the considered system of equations, it is shown that the initial conditions for electric and magnetic potentials cannot be specified independently, as they are uniquely determined by the initial conditions for the components of the displacement vector.
A concrete mixed initial-boundary value problem is considered for a bounded elastic, electromagnetic solid with a single interior crack. The exterior boundary of the solid is divided into two disjoint parts: (i) a Dirichlet part with prescribed displacement vector, electric potential and magnetic potential; (ii) a Neumann part with prescribed mechanical stress vector, normal components of the electric displacement vector and the magnetic induction vector. Moreover, it is assumed that the electric potential and the normal component of the electric displacement vector as well as the magnetic potential and the normal component of the magnetic induction vector are continuous across the crack surface. Initial conditions are given only for the mechanical displacement vector and its time derivative.
Asymptotic expansions of the solution are obtained near the crack edge and at intersections of the lines where the boundary conditions are set.Vibrations of nonlinear elastic lattices: low- and high-frequency dynamic models, internal resonances and modes couplinghttps://zbmath.org/1472.740812021-11-25T18:46:10.358925Z"Andrianov, Igor V."https://zbmath.org/authors/?q=ai:andrianov.igor-v"Danishevskyy, Vladyslav V."https://zbmath.org/authors/?q=ai:danishevsky.vladyslav"Rogerson, Graham"https://zbmath.org/authors/?q=ai:rogerson.graham-aSummary: We aim to study how the interplay between the effects of nonlinearity and heterogeneity can influence on the distribution and localization of energy in discrete lattice-type structures. As the classical example, vibrations of a cubically nonlinear elastic lattice are considered. In contrast with many other authors, who dealt with infinite and periodic lattices, we examine a finite-size model. Supposing the length of the lattice to be much larger than the distance between the particles, continuous macroscopic equations suitable to describe both low- and high-frequency motions are derived. Acoustic and optical vibrations are studied asymptotically by the method of multiple time scales. For numerical simulations, the Runge-Kutta fourth-order method is employed. Internal resonances and energy exchange between the vibrating modes are predicted and analysed. It is shown that the decrease in the number of particles restricts energy transfers to higher-order modes and prevents the equipartition of energy between all degrees of freedom. The conditions for a possible reduction in the original nonlinear system are also discussed.Interfacial acoustic waves in one-dimensional anisotropic phononic bicrystals with a symmetric unit cellhttps://zbmath.org/1472.740822021-11-25T18:46:10.358925Z"Darinskii, A. N."https://zbmath.org/authors/?q=ai:darinskii.a-n"Shuvalov, A. L."https://zbmath.org/authors/?q=ai:shuvalov.alexander-lSummary: The paper is concerned with the interfacial acoustic waves localized at the internal boundary of two different perfectly bonded semi-infinite one-dimensional phononic crystals represented by periodically layered or functionally graded elastic structures. The unit cell is assumed symmetric relative to its midplane, whereas the constituent materials may be of arbitrary anisotropy. The issue of the maximum possible number of interfacial waves per full stop band of a phononic bicrystal is investigated. It is proved that, given a fixed tangential wavenumber, the lowest stop band admits at most one interfacial wave, while an upper stop band admits up to three interfacial waves. The results obtained for the case of generally anisotropic bicrystals are specialized for the case of a symmetric sagittal plane.A proof that multiple waves propagate in ensemble-averaged particulate materialshttps://zbmath.org/1472.741062021-11-25T18:46:10.358925Z"Gower, Artur L."https://zbmath.org/authors/?q=ai:gower.artur-l"Abrahams, I. David"https://zbmath.org/authors/?q=ai:abrahams.i-david"Parnell, William J."https://zbmath.org/authors/?q=ai:parnell.william-jSummary: Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterize wave propagation through an inhomogeneous material, the most crucial parameter is the \textit{effective wavenumber}. For this reason, there are many published studies on how to calculate a single effective wavenumber. Here, we present a proof that there \textit{does not} exist a unique effective wavenumber; instead, there are an infinite number of such (complex) wavenumbers. We show that in most parameter regimes only a small number of these effective wavenumbers make a significant contribution to the wave field. However, to accurately calculate the reflection and transmission coefficients, a large number of the (highly attenuating) effective waves is required. For clarity, we present results for scalar (acoustic) waves for a two-dimensional material filled (over a half-space) with randomly distributed circular cylindrical inclusions. We calculate the effective medium by ensemble averaging over all possible inhomogeneities. The proof is based on the application of the Wiener-Hopf technique and makes no assumption on the wavelength, particle boundary conditions/size or volume fraction. This technique provides a simple formula for the reflection coefficient, which can be explicitly evaluated for monopole scatterers. We compare results with an alternative numerical matching method.Longitudinal wave propagation in a one-dimensional quasi-periodic waveguidehttps://zbmath.org/1472.741092021-11-25T18:46:10.358925Z"Sorokin, Vladislav S."https://zbmath.org/authors/?q=ai:sorokin.vladislav-sSummary: The paper deals with the analysis of wave propagation in a general one-dimensional (1D) non-uniform waveguide featuring multiple modulations of parameters with different, arbitrarily related, spatial periods. The considered quasi-periodic waveguide, in particular, can be viewed as a model of pure periodic structures with imperfections. Effects of such imperfections on the waveguide frequency bandgaps are revealed and described by means of the method of varying amplitudes and the method of direct separation of motions. It is shown that imperfections cannot considerably degrade wave attenuation properties of 1D periodic structures, e.g. reduce widths of their frequency bandgaps. Attenuation levels and frequency bandgaps featured by the quasi-periodic waveguide are studied without imposing any restrictions on the periods of the modulations, e.g. for their ratio to be rational. For the waveguide featuring relatively small modulations with periods that are not close to each other, each of the frequency bandgaps, to the leading order of smallness, is controlled only by one of the modulations. It is shown that introducing additional spatial modulations to a pure periodic structure can enhance its wave attenuation properties, e.g. a relatively low-frequency bandgap can be induced providing vibration attenuation in frequency ranges where damping is less effective.A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materialshttps://zbmath.org/1472.741142021-11-25T18:46:10.358925Z"Nobili, A."https://zbmath.org/authors/?q=ai:nobili.andrea"Radi, E."https://zbmath.org/authors/?q=ai:radi.enrico"Signorini, C."https://zbmath.org/authors/?q=ai:signorini.cesareSummary: Motivated by the unexpected appearance of shear horizontal Rayleigh surface waves, we investigate the mechanics of antiplane wave reflection and propagation in couple stress (CS) elastic materials. Surface waves arise by mode conversion at a free surface, whereby bulk travelling waves trigger inhomogeneous modes. Indeed, Rayleigh waves are perturbations of the travelling mode and stem from its reflection at grazing incidence. As is well known, they correspond to the real zeros of the Rayleigh function. Interestingly, we show that the same generating mechanism sustains a new inhomogeneous wave, corresponding to a purely imaginary zero of the Rayleigh function. This wave emerges from `reflection' of a bulk standing mode: This produces a new type of Rayleigh-like wave that travels \textit{away from}, as opposed to along, the free surface, with a speed lower than that of bulk shear waves. Besides, a third complex zero of the Rayleigh function may exist, which represents waves attenuating/exploding both along and away from the surface. Since none of these zeros correspond to leaky waves, a new classification of the Rayleigh zeros is proposed. Furthermore, we extend to CS elasticity Mindlin's boundary conditions, by which partial waves are identified, whose interference lends Rayleigh-Lamb guided waves. Finally, asymptotic analysis in the thin-plate limit provides equivalent one-dimensional models.A Mathieu function boundary spectral method for scattering by multiple variable poro-elastic plates, with applications to metamaterials and acousticshttps://zbmath.org/1472.741172021-11-25T18:46:10.358925Z"Colbrook, Matthew J."https://zbmath.org/authors/?q=ai:colbrook.matthew-j"Kisil, Anastasia V."https://zbmath.org/authors/?q=ai:kisil.anastasia-vSummary: Many problems in fluid mechanics and acoustics can be modelled by Helmholtz scattering off poro-elastic plates. We develop a boundary spectral method, based on collocation of local Mathieu function expansions, for Helmholtz scattering off multiple variable poro-elastic plates in two dimensions. Such boundary conditions, namely the varying physical parameters and coupled thin-plate equation, present a considerable challenge to current methods. The new method is fast, accurate and flexible, with the ability to compute expansions in thousands (and even tens of thousands) of Mathieu functions, thus making it a favourable method for the considered geometries. Comparisons are made with elastic boundary element methods, where the new method is found to be faster and more accurate. Our solution representation directly provides a sine series approximation of the far-field directivity and can be evaluated near or on the scatterers, meaning that the near field can be computed stably and efficiently. The new method also allows us to examine the effects of varying stiffness along a plate, which is poorly studied due to limitations of other available techniques. We show that a power-law decrease to zero in stiffness parameters gives rise to unexpected scattering and aeroacoustic effects similar to an acoustic black hole metamaterial.Wave scattering on lattice structures involving an array of crackshttps://zbmath.org/1472.741202021-11-25T18:46:10.358925Z"Maurya, Gaurav"https://zbmath.org/authors/?q=ai:maurya.gaurav"Sharma, Basant Lal"https://zbmath.org/authors/?q=ai:sharma.basant-lalSummary: Scattering of waves as a result of a vertical array of equally spaced cracks on a square lattice is studied. The convenience of Floquet periodicity reduces the study to that of scattering of a specific wave-mode from a single crack in a waveguide. The discrete Green's function, for the waveguide, is used to obtain the semi-analytical solution for the scattering problem in the case of finite cracks whereas the limiting case of semi-infinite cracks is tackled by an application of the Wiener-Hopf technique. Reflectance and transmittance of such an array of cracks, in terms of incident wave parameters, is analysed. Potential applications include construction of tunable atomic-scale interfaces to control energy transmission at different frequencies.Scattering on a square lattice from a crack with a damage zonehttps://zbmath.org/1472.741222021-11-25T18:46:10.358925Z"Sharma, Basant Lal"https://zbmath.org/authors/?q=ai:sharma.basant-lal"Mishuris, Gennady"https://zbmath.org/authors/?q=ai:mishuris.gennady-sSummary: A semi-infinite crack in an infinite square lattice is subjected to a wave coming from infinity, thereby leading to its scattering by the crack surfaces. A partially damaged zone ahead of the crack tip is modelled by an arbitrarily distributed stiffness of the damaged links. While an open crack, with an atomically sharp crack tip, in the lattice has been solved in closed form with the help of the scalar
Wiener-Hopf formulation
[the first author, SIAM J. Appl. Math. 75, No. 3, 1171--1192 (2015; Zbl 1322.78010);
SIAM J. Appl. Math. 75, No. 4, 1915--1940 (2015; Zbl 1333.74049)],
the problem considered here becomes very intricate depending on the nature of the damaged links. For instance, in the case of a partially bridged finite zone it involves a \(2 \times 2\) matrix kernel of formidable class. But using an original technique, the problem, including the general case of arbitrarily damaged links, is reduced to a scalar one with the exception that it involves solving an auxiliary linear system of \(N \times N\) equations, where \(N\) defines the length of the damage zone. The proposed method does allow, effectively, the construction of an exact solution. Numerical examples and the asymptotic approximation of the scattered field far away from the crack tip are also presented.Elastic shocks in relativistic rigid rods and ballshttps://zbmath.org/1472.741242021-11-25T18:46:10.358925Z"Costa, João L."https://zbmath.org/authors/?q=ai:costa.joao-lopes"Natário, José"https://zbmath.org/authors/?q=ai:natario.joseSummary: We study the free boundary problem for the `hard phase' material introduced by
\textit{D. Christodoulou} [Arch. Ration. Mech. Anal. 130, No. 4, 343--400 (1995; Zbl 0841.76097)],
both for rods in \((1 + 1)\)-dimensional Minkowski space-time and for spherically symmetric balls in \((3 + 1)\)-dimensional Minkowski space-time. Unlike Christodoulou, we do not consider a `soft phase', and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.The dispersion properties of localized elastic waves in a orthotropic layer between orthotropic half-spaceshttps://zbmath.org/1472.741252021-11-25T18:46:10.358925Z"Glukhov, I. A."https://zbmath.org/authors/?q=ai:glukhov.i-a"Storozhev, V. I."https://zbmath.org/authors/?q=ai:storozhev.v-iSummary: Are presented the obtaining of the numerical-analytical solution for the tree-dimensional problem of the propagation of localized elastic waves along arbitrarily oriented directions in the plane orthotropic deformable layer between two similar orthotropic half-spaces. In the case of waves with symmetrical along layer thickness elastic displacements are obtained and qualitatively analyzed the general dispersion relations. The results of calculations of fragment spectra for waves of the analyzed type for considered waveguide with the components from the real orthotropic materials are obtained. Are characterized the changes in the topology of the spectra caused by varying the orientation of the direction of propagation.Location of eigenmodes of Euler-Bernoulli beam model under fully non-dissipative boundary conditionshttps://zbmath.org/1472.741342021-11-25T18:46:10.358925Z"Shubov, Marianna A."https://zbmath.org/authors/?q=ai:shubov.marianna-aSummary: The Euler-Bernoulli beam model with non-conservative feedback-type boundary conditions is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivative of displacement and slope at the right end. The boundary matrix containing four control parameters relates input and observation. The following results are presented: (i) if one and only one of the control parameters is positive and the rest of them are equal to zero, then the set of the eigenmodes is located in the open left half-plane of the complex plane, which means that all eigenmodes are stable; (ii) if the diagonal elements of the boundary matrix are positive and off-diagonal elements are zeros, then the set of the eigenmodes is located in the open left half-plane, which implies stability of all eigenmodes; (iii) specific combinations of the diagonal and off-diagonal elements have been found to ensure the stability results. To prove the results, two special relations between the eigenmodes and mode shapes of the non-self-adjoint problem and clamped-free self-adjoint problem have been established.The dynamics of thin plates on elastic foundation under the action of local loadshttps://zbmath.org/1472.741402021-11-25T18:46:10.358925Z"Vetrov, O. S."https://zbmath.org/authors/?q=ai:vetrov.o-s"Shevchenko, V. P."https://zbmath.org/authors/?q=ai:shevchenko.volodymyr-pSummary: The fundamental solution of the dynamic equation of an orthotropic plate is constructed by using the method of integral transformations. The problems of the action on the thin plate suddenly applied dynamic and impulse loads are numerically investigated.Stabilization of vibrations of the Kirchhoff plate by using a state feedbackhttps://zbmath.org/1472.741422021-11-25T18:46:10.358925Z"Zuyev, A. L."https://zbmath.org/authors/?q=ai:zuyev.alexander"Novikova, Yu. V."https://zbmath.org/authors/?q=ai:novikova.yu-vSummary: An infinite system of differential equations that describes the vibrations of the Kirchhoff plate is considered. Feedback control functionals, depending on the generalized velocities, are constructed for the system considered. A theorem on the partial asymptotic stability of the equilibrium of the closed-loop system is proved.A frictional contact problem with wear diffusionhttps://zbmath.org/1472.741722021-11-25T18:46:10.358925Z"Kalita, Piotr"https://zbmath.org/authors/?q=ai:kalita.piotr"Szafraniec, Pawel"https://zbmath.org/authors/?q=ai:szafraniec.pawel"Shillor, Meir"https://zbmath.org/authors/?q=ai:shillor.meirThe authors study a model for the dynamic frictional contact between a viscoelastic body and a moving foundation. The contact, described by a generalized compliance condition, includes friction and wear between a viscoelastic body and a reactive foundation. The contact surface is a manifold and the model is related to the wear of mechanical joints and orthopedic biomechanics where the wear debris are trapped. The authors provide the existence of a weak solution by the use a truncation technique and the Kakutani-Ky Fan-Glicksberg fixed point theorem.Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equationshttps://zbmath.org/1472.760032021-11-25T18:46:10.358925Z"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.6"Wu, Wenpei"https://zbmath.org/authors/?q=ai:wu.wenpeiThe paper in question deals with compressible viscoelastic electrical conducting fluids. Their flows are governed by the elastic Navier-Stokes-Poisson system
\begin{align*} \partial_t\rho + \nabla\cdot(\rho u) & = 0,\\
\partial_t(\rho u) + \nabla\cdot (\rho u \otimes u) + \nabla P(\rho) & = \mu\Delta u + (\mu + \lambda)\nabla\nabla\cdot u + c^2 \nabla\cdot(\rho \mathbb F\mathbb F^T) + \rho\nabla\Phi,\\
\partial_t\mathbb F + u \cdot \nabla \mathbb F & = \nabla u \mathbb F,\\
\Delta \Phi & = \rho - \overline\rho, \end{align*}
which holds true on a certain bounded domain \(\Omega\subset \mathbb R^3\). Here the unknowns are the density \(\rho\), the velocity \(u,\) the deformation gradient \(\mathbb F,\) and the electrostatic potential \(\Phi\).
The system is endowed with an initial condition \[(\rho, u, \mathbb F)(x,0) = (\rho_0,u_0,\mathbb F_0)(x),\] with the homogeneous Dirichlet bouundary condition for \(u\) and with either
\[ \Phi|_{\partial\Omega} = 0\qquad \mbox{or }\quad \nabla\Phi\cdot \nu|_{\partial\Omega} = 0. \]
The main theorem of this paper provides the global-in-time existence of a unique global solution. This result is obtained under additional assumption on the smallness of initial data \((\rho_0 - \overline\rho,u_0,\mathbb F_0)\).
In order to obtain the necessary estimates allowing multiple usage of the local-in-time existence theorem, the authors work with the deformation \(\varphi\) rather than with the deformation gradient \(\mathbb F\). Here the deformation is defined as \(\varphi:= X(x,t) - x\), where \(X\) is the Lagrangian coordinate, i.e., an inverse to a function \(x(X,t)\) defined by the following ordinary differential equation
\begin{align*} \frac{\mathrm{d}x(X,t)}{\mathrm{d}t} & = u(x(X,t),t),\\
x(X,0) & = X. \end{align*}On the dynamics of thin layers of viscous flows inside another viscous fluidhttps://zbmath.org/1472.760082021-11-25T18:46:10.358925Z"Pernas Castaño, Tania"https://zbmath.org/authors/?q=ai:pernas-castano.tania"Velázquez, Juan J. L."https://zbmath.org/authors/?q=ai:velazquez.juan-j-lSummary: In this work we will study the dynamics of a thin layer of a viscous fluid which is embedded in the interior of another viscous fluid. The resulting flow can be approximated by means of the solutions of a free boundary problem for the Stokes equation in which one of the unknowns is the shape of a curve which approximates the geometry of the thin layer of fluid. We also derive the equation yielding the thickness of this fluid. This model, that will be termed as the \textit{Geometric Free Boundary Problem}, will be derived using matched asymptotic expansions. We will prove that the Geometric Free Boundary Problem is well posed and the solutions of the thickness equation are well defined (in particular they do not yield breaking of fluid layers) as long as the solutions of the Geometric Free Boundary Problem exist.Trapped modes in a multi-layer fluidhttps://zbmath.org/1472.760142021-11-25T18:46:10.358925Z"Cal, F. S."https://zbmath.org/authors/?q=ai:cal.filipe-s"Dias, G. A. S."https://zbmath.org/authors/?q=ai:dias.goncalo-a-s"Pereira, B. M. M."https://zbmath.org/authors/?q=ai:pereira.b-m-m"Videman, J. H."https://zbmath.org/authors/?q=ai:videman.juha-hansSummary: In this article, we study the existence of solutions for the problem of interaction of linear water waves with an array of three-dimensional fixed structures in a density-stratified multi-layer fluid, where in each layer the density is assumed to be constant. Considering time-harmonic small-amplitude motion, we present recursive formulae for the coefficients of the eigenfunctions of the spectral problem associated with the water-wave problem in the absence of obstacles and for the corresponding dispersion relation. We derive a variational and operator formulation for the problem with obstacles and introduce a sufficient condition for the existence of propagating waves trapped in the vicinity of the array of obstacles. We present several (arrays of) structures supporting trapped waves and discuss the possibility of approximating the continuously stratified fluid by a multi-layer model.Scattering and radiation of water waves by a submerged rigid disc in a two-layer fluidhttps://zbmath.org/1472.760162021-11-25T18:46:10.358925Z"Islam, Najnin"https://zbmath.org/authors/?q=ai:islam.najnin"Kundu, Souvik"https://zbmath.org/authors/?q=ai:kundu.souvik"Gayen, Rupanwita"https://zbmath.org/authors/?q=ai:gayen.rupanwitaSummary: Interaction of water waves with a horizontal rigid disc submerged in the lower layer of a two-layer fluid is studied in three dimensions using linear theory. The governing boundary value problem is reduced to a two-dimensional hypersingular integral equation. This integral equation is further reduced to a one-dimensional Fredholm integral equation of the second kind in terms of a newly defined function. The solution to the latter integral equation is used to compute the total scattering cross section and the hydrodynamic force for the scattering problem and the added mass and the damping coefficient for the radiation problem. Haskind relations connecting the solutions of the radiation and the scattering problems are also derived. The effects of variations of the submergence depth of the disc and the depth of the upper layer on different physical quantities are investigated. We observe amplification of the added mass and the damping coefficient, the total scattering cross section and the hydrodynamic force when the disc goes near the interface or when the height of the upper layer decreases. Known results for a horizontal disc submerged in a single-layer fluid of infinite depth are recovered from the present analysis.Kernel representation of long-wave dynamics on a uniform slopehttps://zbmath.org/1472.760182021-11-25T18:46:10.358925Z"Shimozono, T."https://zbmath.org/authors/?q=ai:shimozono.takenoriSummary: Long-wave propagation on a uniformly sloping beach is formulated as a transient-response problem, with initially stationary water subjected to an incident wave. Both the water surface elevation and the horizontal flow velocity on the slope can be represented as convolutions of the rate of displacement of the water surface at the toe of the slope with a singular kernel function of time and space. The kernel, which is typically expressed in the form of an infinite series, accommodates the dynamic processes of long waves, such as shoaling, reflection, and multiple reflections over the slope and yields exact solutions of the linear shallow water equations for any smooth incident wave. The kernel convolution can be implemented numerically by using double exponential formulas to avoid the kernel singularity. The kernel formulation can be extended readily to nonlinear dynamics \textit{via} the hodograph transform, which in turn enables the instantaneous prediction of nonlinear wave properties and of the occurrence of wave breaking in the near-shore area. This general description of long-wave dynamics provides new insights into the long-studied problem.General rogue wave solutions of the coupled Fokas-Lenells equations and non-recursive Darboux transformationhttps://zbmath.org/1472.760202021-11-25T18:46:10.358925Z"Ye, Yanlin"https://zbmath.org/authors/?q=ai:ye.yanlin"Zhou, Yi"https://zbmath.org/authors/?q=ai:zhou.yi"Chen, Shihua"https://zbmath.org/authors/?q=ai:chen.shihua"Baronio, Fabio"https://zbmath.org/authors/?q=ai:baronio.fabio"Grelu, Philippe"https://zbmath.org/authors/?q=ai:grelu.philippeSummary: We formulate a non-recursive Darboux transformation technique to obtain the general \(n\) th-order rational rogue wave solutions to the coupled Fokas-Lenells system, which is an integrable extension of the noted Manakov system, by considering both the double-root and triple-root situations of the spectral characteristic equation. Based on the explicit fundamental and second-order rogue wave solutions, we demonstrate several interesting rogue wave dynamics, among which are coexisting rogue waves and anomalous Peregrine solitons. Our solutions are generalized to include the complete background-field parameters and therefore helpful for future experimental study.A fluid mechanic's analysis of the teacup singularityhttps://zbmath.org/1472.760212021-11-25T18:46:10.358925Z"Barkley, Dwight"https://zbmath.org/authors/?q=ai:barkley.dwightSummary: The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with the swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. The linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and from the swirl, and enforcing incompressibility and enforcing flow confinement. The key pressure field driving the blowup of velocity gradients is that confining the fluid within the cylinder walls. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model captures key features in the mechanics of the blowup scenario.Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torushttps://zbmath.org/1472.760232021-11-25T18:46:10.358925Z"Sakajo, Takashi"https://zbmath.org/authors/?q=ai:sakajo.takashiSummary: A steady solution of the incompressible Euler equation on a toroidal surface \(\mathbb{T}_{R , r}\) of major radius \(R\) and minor radius \(r\) is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, \( \nabla_{\mathbb{T}_{R , r}}^2 \psi = c \text{e}^{d \psi} +( 8 / d) \kappa \), where \(\nabla_{\mathbb{T}_{R , r}}^2\) and \(\kappa\) denote the Laplace-Beltrami operator and the Gauss curvature of the toroidal surface respectively, and \(c, d\) are real parameters with \textit{cd} < 0. This is a generalization of the flows with smooth vorticity distributions owing to
[\textit{J. T. Stuart}, J. Fluid Mech. 29, 417--440 (1967; Zbl 0152.45403)]
in the plane and
[\textit{D. G. Crowdy}, J. Fluid Mech. 498, 381--402 (2004; Zbl 1059.76012)]
on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio \(\alpha = R/r\). A comparison with the Stuart vortex on the flat torus is also made.Desingularization of multiscale solutions to planar incompressible Euler equationshttps://zbmath.org/1472.760242021-11-25T18:46:10.358925Z"Wan, Jie"https://zbmath.org/authors/?q=ai:wan.jieSummary: In this paper, we consider the desingularization of multiscale solutions to 2D steady incompressible Euler equations. When the background flow \(\psi_0\) is nontrivial, we construct a family of solutions which has nonzero vorticity in small neighborhoods of a given collection of points. One prescribed set of points comprises minimizers of the Kirchhoff-Routh function, while another part of points is on the boundary determined by both \(\psi_0\) and Green's function. Moreover, heights and circulation of solutions have two kinds of scale. We prove the results by considering maximization problem for the vorticity and analyzing the asymptotic behavior of the maximizers.Stokes and Navier-Stokes equations subject to partial slip on uniform \(C^{2,1}\)-domains in \(L_q\)-spaceshttps://zbmath.org/1472.760302021-11-25T18:46:10.358925Z"Hobus, Pascal"https://zbmath.org/authors/?q=ai:hobus.pascal"Saal, Jürgen"https://zbmath.org/authors/?q=ai:saal.jurgenThe article presents a comprehensive investigation of the well-posedness of the Stokes and Navier-Stokes initial-value problems subject to certain partial slip boundary conditions on uniform \(C^{2,1}\)-domains, including a class of non-Helmholtz domains, that is, domains where \(L_q\) is not equal to the direct sum of solenoidal fields and gradient fields. For this purpose, the authors introduce a generalized version of the Helmholtz decomposition, which exists for a large class of domains. Moreover, its existence is shown to be necessary and sufficient for the well-posedness of the Stokes resolvent problem under consideration.
The very detailed analysis is based on the resolvent problem of the heat equation subject to perfect slip boundary conditions, which is treated by a localization procedure and perturbation arguments. Since the associated resolvent operator commutes with the generalized Helmholtz decomposition, its properties can be transferred to the Stokes resolvent problem in the case of perfect slip. Subsequently, the more general partial slip boundary conditions are treated by a Neumann series argument. The authors conclude that the associated Stokes operator generates an analytic semigroup, and they derive suitable \(L_p\)-\(L_q\) estimates. Finally, the contraction mapping principle leads to the existence of local-in-time mild solutions to the associated Navier-Stokes problem.Spatial decay of the vorticity field of time-periodic viscous flow past a bodyhttps://zbmath.org/1472.760342021-11-25T18:46:10.358925Z"Eiter, Thomas"https://zbmath.org/authors/?q=ai:eiter.thomas|eiter.thomas-walter"Galdi, Giovanni P."https://zbmath.org/authors/?q=ai:galdi.giovanni-paoloThe three-dimensional exterior problem for the Navier-Stokes equations is considered for time-periodic flows. The exponential decay of the vorticity outside the wake region is shown, uniformly in time. Moreover, the time fluctuation of the vorticity in the wake region decays algebraically and faster that the vorticity itself. This means that far from the body, the vorticity field behaves like that of the steady state exterior problem.Fluctuation theorem and extended thermodynamics of turbulencehttps://zbmath.org/1472.760462021-11-25T18:46:10.358925Z"Porporato, Amilcare"https://zbmath.org/authors/?q=ai:porporato.amilcare"Hooshyar, Milad"https://zbmath.org/authors/?q=ai:hooshyar.milad"Bragg, Andrew D."https://zbmath.org/authors/?q=ai:bragg.andrew-d"Katul, Gabriel"https://zbmath.org/authors/?q=ai:katul.gabriel-gSummary: Turbulent flows are out-of-equilibrium because the energy supply at large scales and its dissipation by viscosity at small scales create a net transfer of energy among all scales. This energy cascade is modelled by approximating the spectral energy balance with a nonlinear Fokker-Planck equation consistent with accepted phenomenological theories of turbulence. The steady-state contributions of the drift and diffusion in the corresponding Langevin equation, combined with the killing term associated with the dissipation, induce a stochastic energy transfer across wavenumbers. The fluctuation theorem is shown to describe the scale-wise statistics of forward and backward energy transfer and their connection to irreversibility and entropy production. The ensuing turbulence entropy is used to formulate an extended turbulence thermodynamics.Homogenization of a micropolar fluid past a porous media with nonzero spin boundary conditionhttps://zbmath.org/1472.760702021-11-25T18:46:10.358925Z"Suárez-Grau, Francisco J."https://zbmath.org/authors/?q=ai:suarez-grau.francisco-javierAfter studying the well-posedness of the microscopic problem -- a micropolar fluid past a perforated domain with non vanishing spin boundary conditions, the author uses the periodic unfolding technique to pass to the periodic homogenization limit. The weak formulation of the upscaled limit is provided.Oriented suspension mechanics with application to improving flow linear dichroism spectroscopyhttps://zbmath.org/1472.761092021-11-25T18:46:10.358925Z"Cupples, G."https://zbmath.org/authors/?q=ai:cupples.g"Smith, D. J."https://zbmath.org/authors/?q=ai:smith.dealton-j|smith.david-j|smith.douglas-j|smith.derek-j"Hicks, M. R."https://zbmath.org/authors/?q=ai:hicks.m-r"Dyson, R. J."https://zbmath.org/authors/?q=ai:dyson.rosemary-jSummary: Flow linear dichroism is a biophysical spectroscopic technique that exploits the shear-induced alignment of elongated particles in suspension. Motivated by the broad aim of optimizing the sensitivity of this technique, and more specifically by a hand-held synthetic biotechnology prototype for waterborne-pathogen detection, a model of steady and oscillating pressure-driven channel flow and orientation dynamics of a suspension of slender microscopic fibres is developed. The model couples the Fokker-Planck equation for Brownian suspensions with the narrow channel flow equations, the latter modified to incorporate mechanical anisotropy induced by the particles. The linear dichroism signal is estimated through integrating the perpendicular components of the distribution function via an appropriate formula which takes the biaxial nature of the orientation into account. For the specific application of pathogen detection via binding of M13 bacteriophage, it is found that increases in the channel depth are more significant in improving the linear dichroism signal than increases in the channel width. Increasing the channel depth to 2 mm and pressure gradient to \(5 \times 10^4 Pa m^{-1}\) essentially maximizes the alignment. Oscillating flow can produce nearly equal alignment to steady flow at appropriate frequencies, which has significant potential practical value in the analysis of small sample volumes.Loss-less propagation, elastic and inelastic interaction of electromagnetic soliton in an anisotropic ferromagnetic nanowirehttps://zbmath.org/1472.780112021-11-25T18:46:10.358925Z"Senthil Kumar, V."https://zbmath.org/authors/?q=ai:kumar.v-senthil"Kavitha, L."https://zbmath.org/authors/?q=ai:kavitha.louis"Boopathy, C."https://zbmath.org/authors/?q=ai:boopathy.c"Gopi, D."https://zbmath.org/authors/?q=ai:gopi.dSummary: Nonlinear interaction of electromagnetic solitons leads to a plethora of interesting physical phenomena in the diverse area of science that include magneto-optics based data storage industry. We investigate the nonlinear magnetization dynamics of a one-dimensional anisotropic ferromagnetic nanowire. The famous Landau-Lifshitz-Gilbert equation (LLG) describes the magnetization dynamics of the ferromagnetic nanowire and the Maxwell's equations govern the propagation dynamics of electromagnetic wave passing through the axis of the nanowire. We perform a uniform expansion of magnetization and magnetic field along the direction of propagation of electromagnetic wave in the framework of reductive perturbation method. The excitation of magnetization of the nanowire is restricted to the normal plane at the lowest order of perturbation and goes out of plane for higher orders. The dynamics of the ferromagnetic nanowire is governed by the modified Korteweg-de Vries (mKdV) equation and the perturbed modified Korteweg-de Vries (pmKdV) equation for the lower and higher values of damping respectively. We invoke the Hirota bilinearization procedure to mKdV and pmKdV equation to construct the multi-soliton solutions, and explicitly analyze the nature of collision phenomena of the co-propagating EM solitons for the above mentioned lower and higher values of Gilbert-damping due to the precessional motion of the ferromagnetic spin. The EM solitons appearing in the higher damping regime exhibit elastic collision thus yielding the fascinating state restoration property, whereas those of lower damping regime exhibit inelastic collision yielding the solitons of suppressed intensity profiles. The propagation of EM soliton in the nanoscale magnetic wire has potential technological applications in optimizing the magnetic storage devices and magneto-electronics.The general coupled Hirota equations: modulational instability and higher-order vector rogue wave and multi-dark soliton structureshttps://zbmath.org/1472.780202021-11-25T18:46:10.358925Z"Zhang, Guoqiang"https://zbmath.org/authors/?q=ai:zhang.guoqiang"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenya"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.5|wang.li.6|wang.li|wang.li.3|wang.li.2|wang.li.1|wang.li.4Summary: The general coupled Hirota equations are investigated, which describe the wave propagations of two ultrashort optical fields in a fibre. Firstly, we study the modulational instability for the focusing, defocusing and mixed cases. Secondly, we present a unified formula of high-order rational rogue waves (RWs) for the focusing, defocusing and mixed cases, and find that the distribution patterns for novel vector rational RWs of focusing case are more abundant than ones in the scalar model. Thirdly, the \(N\) th-order vector semirational RWs can demonstrate the coexistence of \(N\) th-order vector rational RWs and \(N\) breathers. Fourthly, we derive the multi-dark-dark solitons for the defocsuing and mixed cases. Finally, we derive a formula for the coexistence of dark solitons and RWs. These results further enrich and deepen the understanding of localized wave excitations and applications in vector nonlinear wave systems.On possible applications of media described by fractional-order models in electromagnetic cloakinghttps://zbmath.org/1472.780222021-11-25T18:46:10.358925Z"Stefański, Tomasz P."https://zbmath.org/authors/?q=ai:stefanski.tomasz-pAuthor's abstract: The purpose of this paper is to open a scientific discussion on possible applications of media described by fractional-order (FO) models (FOMs) in electromagnetic cloaking. A 2-D cloak based on active sources and the surface equivalence theorem is simulated. It employs a medium described by FOM in communication with sources cancelling the scattered field. A perfect electromagnetic active cloak is thereby demonstrated with the use of a finite-difference time-domain method combined with a simulation algorithm of non-monochromatic wave propagation in the media described by FOM. The application of constitutive relations based on FOMs in Maxwell's equations provides solutions which correspond to the results reported for the time-fractional diffusion-wave equation, which is non-relativistic, like the classical diffusion equation. This property is employed in the presented cloaking scheme for communication with active current sources around the cloak, which cancel the scattered field of an object inside the cloak. Although in the real world perfect invisibility is impossible to obtain due to the constraint of light speed, it is possible to obtain a perfect cloak in theoretical considerations by using FO formulation of electro-magnetism. It is worth noticing that numerous literature sources experimentally confirm the existence of electromagnetic media described by FOMs; hence, the presented numerical results should hopefully stimulate further investigations related to applications of FOMs in electromagnetic cloaking.Numerical simulation of formation and evolution of dissipative breathers of the classical Heisenberg antiferromagnet modelhttps://zbmath.org/1472.780302021-11-25T18:46:10.358925Z"Muminov, Kh. Kh."https://zbmath.org/authors/?q=ai:muminov.kh-kh"Muhamedova, Sh. F."https://zbmath.org/authors/?q=ai:muhamedova.sh-fSummary: In the present paper we conduct numerical simulation of the breather (soliton-like) solutions of classical Heisenberg antiferromagnetic acted upon by the variable external electromagnetic fields pumping and dissipation. The numerical recipe of simulation on the basic of stereographic projection is suggested avoiding singularities on the poles of the Bloch sphere. Parameters of regime of formation of dissipative breathers are determined.Laplace transform approach for the dynamics of \(N\) qubits coupled to a resonatorhttps://zbmath.org/1472.810662021-11-25T18:46:10.358925Z"Amico, Mirko"https://zbmath.org/authors/?q=ai:amico.mirko"Berman, Oleg L."https://zbmath.org/authors/?q=ai:berman.oleg-l"Kezerashvili, Roman Ya."https://zbmath.org/authors/?q=ai:kezerashvili.roman-yaSummary: An approach to use the method of Laplace transform for the perturbative solution of the Schrödinger equation at any order of the perturbation for a system of \(N\) qubits coupled to a cavity with \(n\) photons is suggested. We investigate the dynamics of a system of \(N\) superconducting qubits coupled to a common resonator with time-dependent coupling. To account for the contribution of the dynamical Lamb effect to the probability of excitation of the qubit, we consider counter-rotating terms in the qubit-photon interaction Hamiltonian. As an example, we illustrate the method for the case of two qubits coupled to a common cavity. The perturbative solutions for the probability of excitation of the qubit show excellent agreement with the numerical calculations.Riccati-type pseudo-potentials, conservation laws and solitons of deformed sine-Gordon modelshttps://zbmath.org/1472.810672021-11-25T18:46:10.358925Z"Blas, H."https://zbmath.org/authors/?q=ai:blas.harold"Callisaya, H. F."https://zbmath.org/authors/?q=ai:callisaya.hector-flores"Campos, J. P. R."https://zbmath.org/authors/?q=ai:campos.j-p-rSummary: Deformed sine-Gordon (DSG) models \(\partial_\xi \partial_\eta w + \frac{d}{d w} V(w) = 0\), with \(V(w)\) being the deformed potential, are considered in the context of the Riccati-type pseudo-potential approach. A compatibility condition of the deformed system of Riccati-type equations reproduces the equation of motion of the DSG models. Then, we provide a pair of linear systems of equations for the DSG model and an associated infinite tower of non-local conservation laws. Through a direct construction and supported by numerical simulations of soliton scatterings, we show that the DSG models, which have recently been defined as quasi-integrable in the anomalous zero-curvature approach [\textit{L. A. Ferreira} and \textit{W. J. Zakrzewski}, J. High Energy Phys. 2011, No. 5, Paper No. 130, 39 p. (2011; Zbl 1296.81035)], possess new towers of infinite number of quasi-conservation laws. We compute numerically the first sets of non-trivial and independent charges (beyond energy and momentum) of the DSG model: the two third order conserved charges and the two fifth order asymptotically conserved charges in thepseudo-potential approach, and the first four anomalies of the new towers of charges, resp ectively. We consider kink-kink, kink-antikink and breather configurations for the Bazeia et al. potential \(V_q(w) = \frac{64}{q^2} \tan^2 \frac{w}{2}(1 - | \sin \frac{w}{2} |^q)^2\) \((q \in \mathbb{R})\), which contains the usual SG potential \(V_2(w) = 2 [1 - \cos(2w)]\). The numerical simulations are performed using the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.Lax connection and conserved quantities of quadratic mean field gameshttps://zbmath.org/1472.810682021-11-25T18:46:10.358925Z"Bonnemain, Thibault"https://zbmath.org/authors/?q=ai:bonnemain.thibault"Gobron, Thierry"https://zbmath.org/authors/?q=ai:gobron.thierry"Ullmo, Denis"https://zbmath.org/authors/?q=ai:ullmo.denisSummary: Mean field games is a new field developed simultaneously in applied mathematics and engineering in order to deal with the dynamics of a large number of controlled agents or objects in interaction. For a large class of these models, there exists a deep relationship between the associated system of equations and the non-linear Schrödinger equation, which allows us to get new insights into the structure of their solutions. In this work, we deal with the related aspects of integrability for such systems, exhibiting in some cases a full hierarchy of conserved quantities and bringing some new questions that arise in this specific context.
{\copyright 2021 American Institute of Physics}On the spectrum and eigenfunctions of the equivariant general boundary value problem outside the sphere for the Schrödinger operator with Coulomb potentialhttps://zbmath.org/1472.810712021-11-25T18:46:10.358925Z"Burskii, V. P."https://zbmath.org/authors/?q=ai:burskii.vladimir-p"Zaretskaya, A. A."https://zbmath.org/authors/?q=ai:zaretskaya.a-aSummary: We consider the Schrödinger equation of hydrogen-type atom with Coulomb potential. The eigenvalue and eigenfunction are found in the case of swing-invariant boundary value problem.Dynamics of solutions of a fractional NLS system with quadratic interactionhttps://zbmath.org/1472.810732021-11-25T18:46:10.358925Z"Esfahani, Amin"https://zbmath.org/authors/?q=ai:esfahani.aminSummary: The purpose of this article is twofold. First, we study the existence and asymptotic behavior of ground states of a fractional Schrodinger system with quadratic interaction. Second, we give some conditions, in terms of the mass and energy of the ground states, under which the solutions of the associated initial value problem have the uniform bound or may blow up in finite time. As a corollary, we show the strong instability of the ground states.
{\copyright 2021 American Institute of Physics}Limiting absorption principle and radiation condition for repulsive Hamiltonianshttps://zbmath.org/1472.810752021-11-25T18:46:10.358925Z"Itakura, Kyohei"https://zbmath.org/authors/?q=ai:itakura.kyoheiSummary: For spherically symmetric repulsive Hamiltonians we prove the limiting absorption principle bound, the radiation condition bounds and the limiting absorption principle. The Sommerfeld uniqueness result also follows as a corollary of these. In particular, the Hamiltonians considered in this paper cover the case of inverted harmonic oscillator. In the proofs of our theorems, we mainly use a commutator argument invented recently by Ito and Skibsted. This argument is simple and elementary, and dose not employ energy cut-offs or the microlocal analysis.Markovian embedding procedures for non-Markovian stochastic Schrödinger equationshttps://zbmath.org/1472.810812021-11-25T18:46:10.358925Z"Li, Xiantao"https://zbmath.org/authors/?q=ai:li.xiantaoSummary: We present embedding procedures for the non-Markovian stochastic Schrödinger equations, arising from studies of quantum systems coupled with bath environments. By introducing auxiliary wave functions, it is demonstrated that the non-Markovian dynamics can be embedded in extended, but Markovian, stochastic models. Two embedding procedures are presented. The first method leads to nonlinear stochastic equations, the implementation of which is much more efficient than the non-Markovian stochastic Schrödinger equations. The stochastic Schrödinger equations obtained from the second procedure involve more auxiliary wave functions, but the equations are linear, and a closed-form generalized quantum master equation for the density-matrix can be obtained. The accuracy of the embedded models is ensured by fitting to the power spectrum. The stochastic force is represented using a linear superposition of Ornstein-Uhlenbeck processes, which are incorporated as multiplicative noise in the auxiliary Schrödinger equations. It is shown that the asymptotic behavior of the spectral density in the low frequency regime, which is responsible for the long-time behavior of the quantum dynamics, can be preserved by using correlated stochastic processes. The approximations are verified by using a spin-boson system as a test example.Gauge transformations of spectral triples with twisted real structureshttps://zbmath.org/1472.810822021-11-25T18:46:10.358925Z"Magee, Adam M."https://zbmath.org/authors/?q=ai:magee.adam-m"Dąbrowski, Ludwik"https://zbmath.org/authors/?q=ai:dabrowski.ludwikSummary: Twisted real structures are well-motivated as a way to implement the conformal transformation of a Dirac operator for a real spectral triple without needing to twist the noncommutative one-forms. We study the coupling of spectral triples with twisted real structures to gauge fields, adopting Morita equivalence via modules and bimodules as a guiding principle and paying special attention to modifications to the inner fluctuations of the Dirac operator. In particular, we analyze the twisted first-order condition as a possible alternative to abandoning the first-order condition in order to go beyond the standard model and elaborate upon the special case of gauge transformations accordingly. Applying the formalism to a toy model, we argue that under certain physically motivated assumptions, the spectral triple based on the left-right symmetric algebra should reduce to that of the standard model of fundamental particles and interactions, as in the untwisted case.
{\copyright 2021 American Institute of Physics}Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein-Gordon equation and numerical comparisonshttps://zbmath.org/1472.810832021-11-25T18:46:10.358925Z"Muda, Y."https://zbmath.org/authors/?q=ai:muda.yuslenita"Akbar, F. T."https://zbmath.org/authors/?q=ai:akbar.fiki-taufik"Kusdiantara, R."https://zbmath.org/authors/?q=ai:kusdiantara.rudy"Gunara, B. E."https://zbmath.org/authors/?q=ai:gunara.bobby-eka"Susanto, H."https://zbmath.org/authors/?q=ai:susanto.hadiSummary: We consider a damped, parametrically driven discrete nonlinear Klein-Gordon equation, that models coupled pendula and micromechanical arrays, among others. To study the equation, one usually uses a small-amplitude wave ansatz, that reduces the equation into a discrete nonlinear Schrödinger equation with damping and parametric drive. Here, we justify the approximation by looking for the error bound with the method of energy estimates. Furthermore, we prove the local and global existence of solutions to the discrete nonlinear Schrödinger equation. To illustrate the main results, we consider numerical simulations showing the dynamics of errors made by the discrete nonlinear equation. We consider two types of initial conditions, with one of them being a discrete soliton of the nonlinear Schrödinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein-Gordon equation.Characterization of Darboux transformations for quantum systems with quadratically energy-dependent potentialshttps://zbmath.org/1472.810852021-11-25T18:46:10.358925Z"Schulze-Halberg, Axel"https://zbmath.org/authors/?q=ai:schulze-halberg.axelSummary: We construct three classes of higher-order Darboux transformations for Schrödinger equations with quadratically energy-dependent potentials by means of generalized Wronskian determinants. Particular even-order cases reduce to the Darboux transformation for conventional (energy-independent) potentials. Our construction is based on an adaptation of the results for coupled Korteweg-de Vries equations [\textit{S. B. Leble} and \textit{N. V. Ustinov}, J. Math. Phys. 34, No. 4, 1421--1428 (1993; Zbl 0774.35075)].
{\copyright 2021 American Institute of Physics}Dirac particle with memory: proper time non-localityhttps://zbmath.org/1472.810872021-11-25T18:46:10.358925Z"Tarasov, Vasily E."https://zbmath.org/authors/?q=ai:tarasov.vasily-eSummary: A generalization of the standard model of Dirac particle in external electromagnetic field is proposed. In the generalization we take into account interactions of this particle with environment, which is described by the memory function. This function takes into account that the behavior of the particle at proper time can depend not only at the present time, but also on the history of changes on finite time interval. In this case the Dirac particle can be considered an open quantum system with non-Markovian dynamics. The violation of the semigroup property of dynamic maps is a characteristic property of dynamics with memory. We use the Fock-Schwinger proper time method and derivatives of non-integer orders with respect to proper time. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described.On the derivative nonlinear Schrödinger equation on the half line with Robin boundary conditionhttps://zbmath.org/1472.810882021-11-25T18:46:10.358925Z"Van Tin, Phan"https://zbmath.org/authors/?q=ai:van-tin.phanSummary: We consider the Schrödinger equation with a nonlinear derivative term on [0, +\(\infty)\) under the Robin boundary condition at 0. Using a virial argument, we obtain the existence of blowing up solutions, and using variational techniques, we obtain stability and instability by blow-up results for standing waves.
{\copyright 2021 American Institute of Physics}Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with \(\mathcal{PT}\)-symmetric harmonic potential via deep learninghttps://zbmath.org/1472.810892021-11-25T18:46:10.358925Z"Zhou, Zijian"https://zbmath.org/authors/?q=ai:zhou.zijian"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenyaSummary: In this paper, we investigate the logarithmic nonlinear Schrödinger (LNLS) equation with the parity-time \((\mathcal{PT})\)-symmetric harmonic potential, which is an important physical model in many fields such as nuclear physics, quantum optics, magma transport phenomena, and effective quantum gravity. Three types of initial value conditions and periodic boundary conditions are chosen to solve the LNLS equation with \(\mathcal{PT}\)-symmetric harmonic potential via the physics-informed neural networks (PINNs) deep learning method, and these obtained results are compared with ones deduced from the Fourier spectral method. Moreover, we also investigate the effectiveness of the PINNs deep learning for the LNLS equation with \(\mathcal{PT}\) symmetric potential by choosing the distinct space widths or distinct optimized steps. Finally, we use the PINNs deep learning method to effectively tackle the data-driven discovery of the LNLS equation with \(\mathcal{PT} \)-symmetric harmonic potential such that the coefficients of dispersion and nonlinear terms or the amplitudes of \(\mathcal{PT}\)-symmetric harmonic potential can be approximately found.A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalitieshttps://zbmath.org/1472.810902021-11-25T18:46:10.358925Z"Antunes, Pedro R. S."https://zbmath.org/authors/?q=ai:antunes.pedro-ricardo-simao"Benguria, Rafael D."https://zbmath.org/authors/?q=ai:benguria.rafael-d"Lotoreichik, Vladimir"https://zbmath.org/authors/?q=ai:lotoreichik.vladimir"Ourmières-Bonafos, Thomas"https://zbmath.org/authors/?q=ai:ourmieres-bonafos.thomaslet \(\Omega \subset {\mathbb R}^2\) be a \(C^\infty\) simply connected domain and let \(n = (n_1,n_2)^\top\) be the outward pointing normal field on \(\partial\Omega\). The Dirac operator with infinite mass boundary conditions in \(L^2(\Omega,{\mathbb C}^2)\) is defined as \[D^\Omega := \begin{pmatrix} 0 & -2\mathrm{i}\partial_z\\
-2\mathrm{i}\partial_{\bar z} & 0 \end{pmatrix}, \] with domain \(\{ u = (u_1,u_2)^\top \in H^1(\Omega,{\mathbb C}^2) : u_2 = \mathrm{i} \mathbf{n}u_1 \text{ on }\partial\Omega \},\) where \(\mathbf{n} := n_1 + \mathrm{i} n_2\) and \(\partial_z, \partial_{\bar{z}}\) are the Wirtinger operators. The spectrum of \(D^\Omega\) is symmetric with respect to the origin and constituted of eigenvalues of finite multiplicity \[ \cdots \leq -E_k(\Omega) \leq\cdots \leq-E_{1}(\Omega) < 0 < E_{1}(\Omega) \leq \cdots \leq E_k(\Omega) \leq \cdots.\] The authors prove the following estimate \[E_1(\Omega) \leq \frac{|\partial\Omega|}{(\pi r_i^2 + |\Omega|)}E_1({\mathbb D}) \] with equality if and only if \(\Omega\) is a disk, where \(r_i\) is the inradius of \(\Omega\) and \(\mathbb D\) is the unit disk. \par The second main result of this paper is the following non-linear variational characterization of \(E_1(\Omega)\). \(E > 0\) is the first non-negative eigenvalue of \(D^\Omega\) if and only if \(\mu^\Omega(E) = 0\), where \[\mu^\Omega(E) := \inf\limits_{u} \frac{4 \int_\Omega |\partial_{\bar z} u|^2 dx - E^2 \int_{\Omega}|u|^2dx + E \int_{\partial\Omega} |u|^2 ds}{\int_\Omega |u|^2 dx}.\] \par The authors propose the following conjecture \[\mu^\Omega(E) \geq \frac{\pi}{|\Omega|}\mu^{\mathbb D}\Big(\sqrt{\frac{|\Omega|}{\pi}}E\Big), \forall E>0\] and provide numerical evidences supporting it. This conjecture implies the validity of the Faber-Krahn-type inequality \(E_1(\Omega) \geq \sqrt{\frac{\pi}{|\Omega|}} E_1({\mathbb D})\) (it is still an open question).A new spectral analysis of stationary random Schrödinger operatorshttps://zbmath.org/1472.810922021-11-25T18:46:10.358925Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Shirley, Christopher"https://zbmath.org/authors/?q=ai:shirley.christopherThe authors consider random Schrödinger operators of the form
\[
-\Delta + \lambda V_{\omega}
\]
and the associated Schrödinger equation, where \(V_{\omega}\) is a realization of a stationary random potential \(V\). The regime under consideration here is \(0<\lambda \ll 1\). The main goal of the authors is to develop a spectral approach to describe the long time behavior of the system beyond perturbative timescales by using ideas from Malliavin calculus, leading to rigorous Mourre type results. In particular, the authors describe the dynamics by a fibered family of spectral perturbation problems. They then state a number of exact resonance conjectures which would require that Bloch waves exist as resonant modes. An approximate resonance result is obtained and the first spectral proof of the decay of time correlations on the kinetic timescale is also provided.Tosio Kato's work on non-relativistic quantum mechanics. IIhttps://zbmath.org/1472.810942021-11-25T18:46:10.358925Z"Simon, Barry"https://zbmath.org/authors/?q=ai:simon.barry.1The work is the second part of a review to Kato's work on nonrelativistic quantum mechanics. It focuses on bounds on the number of eigenvalues of the helium atom, on the absence of embedded bound states, on scattering theory under a trace class condition, Kato smoothness, the adiabatic theorem, and the Trotter product formula.
The author is known for the clarity of his presentation which is reflected in this work as well. The review can also serve as an introduction of the subject, since the results are not merely reviewed but put in a current perspective of the field. An example of this is the appendix where the inequality \(|p|>2/(\pi |x|)\) in \(d=3\), known as Kato's inequality or Herbst inequality, is treated. It is put in the context of the groundstate transform which [\textit{R. L. Frank} et al., J. Am. Math. Soc. 21, No. 4, 925--950 (2008; Zbl 1202.35146)] used to prove a generalization for fractional powers of \(p:=-i\nabla\).
For Part I see the author [Bull. Math. Sci. 8, No. 1, 121--232 (2018; Zbl 1416.81063)]Fock quantization of canonical transformations and semiclassical asymptotics for degenerate problemshttps://zbmath.org/1472.810992021-11-25T18:46:10.358925Z"Dobrokhotov, Sergei"https://zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu"Nazaikinskii, Vladimir"https://zbmath.org/authors/?q=ai:nazaikinskii.vladimir-eSummary: The aim of this work is to explain the role played by the Fock quantization of canonical transformations in the construction of the global semiclassical (high-frequency) asymptotic approximation. This role may well pass unnoticed as long as one deals with nondegenerate differential equations. However, the situation is different for some classes of equations with degeneration, where the Fock quantization of canonical transformations becomes instrumental in the construction of asymptotic solutions.
For the entire collection see [Zbl 1472.53006].Remarks on semiclassical wavefront sethttps://zbmath.org/1472.811002021-11-25T18:46:10.358925Z"Kameoka, Kentaro"https://zbmath.org/authors/?q=ai:kameoka.kentaroSummary: The essential support of the symbol of a semiclassical pseudodifferential operator is characterized by semiclassical wavefront sets of distributions. The proof employs a coherent state whose center in phase space is dependent on Planck's constant.Reduction and coherent stateshttps://zbmath.org/1472.811012021-11-25T18:46:10.358925Z"Rousseva, Jenia"https://zbmath.org/authors/?q=ai:rousseva.jenia"Uribe, Alejandro"https://zbmath.org/authors/?q=ai:uribe.alejandroThe authors apply a quantum version of dimensional reduction to Gaussian coherent states in Bargmann space to obtain squezed states on complex projective spaces (Definition 1.13) which lay perfectly in semiclassical approximation. Besides they are governed by a symbol calculus. The authors prove semiclassical norm estimates and a propagation result. The article includes the following topics: reduction of Gausian coherent states, squezed spin coherent states, covariance and Gaussian states, symbols, quantized Kahler manifolds, Bargmann spaces, reduction, classical propagation, quantum propagation, examples.Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random modelshttps://zbmath.org/1472.811042021-11-25T18:46:10.358925Z"Hislop, Peter D."https://zbmath.org/authors/?q=ai:hislop.peter-d"Marx, Christoph A."https://zbmath.org/authors/?q=ai:marx.christoph-aSummary: We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrödinger operators on infinite graphs. While previous results were limited to random Schrödinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrödinger operators independent of the specific nature of the potential. Following ideas by Bourgain and Klein, we consider the density of states outer measure (DOSoM), which is well defined for all (deterministic) Schrödinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the \(\ell^\infty \)-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrödinger operators on \(\mathbb{Z}^d\), this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrödinger operators on the Bethe lattice, we obtain log-Hölder dependence of the DOSoM on the potential. As an important consequence of our deterministic framework, we obtain a modulus of continuity for the density of states measure (DOSm) of ergodic Schrödinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrödinger operators on the dependence of the DOSm on the single-site probability measure by formulating this problem in the ergodic framework using the quantile function associated with the random potential.Configurational complexity of nonautonomous discrete one-soliton and rogue waves in Ablowitz-Ladik-Hirota waveguidehttps://zbmath.org/1472.811062021-11-25T18:46:10.358925Z"Thakur, Pooja"https://zbmath.org/authors/?q=ai:thakur.pooja"Gleiser, Marcelo"https://zbmath.org/authors/?q=ai:gleiser.marcelo"Kumar, Anil"https://zbmath.org/authors/?q=ai:kumar.anil"Gupta, Rama"https://zbmath.org/authors/?q=ai:gupta.ramaSummary: We compute the configurational complexity (CC) for discrete soliton and rogue waves traveling along an Ablowitz-Ladik-Hirota (ALH) waveguide and modeled by a discrete nonlinear Schrödinger equation. We show that for a specific range of the soliton transverse direction \(\kappa\) propagating along the parametric time \(\zeta(t)\), CC reaches an evolving series of global minima. These minima represent maximum compressibility of information in the momentum modes along the Ablowitz-Ladik-Hirota waveguide. Computing the CC for rogue waves as a function of background amplitude modulation \(\mu\), we show that it displays two essential features: a maximum representing the optimal value for the rogue wave inception (the ``gradient catastrophe'') and saturation representing the rogue wave dispersion into constituent wave modes. We show that saturation is achieved earlier for higher values of modulation amplitude as the discrete rogue wave evolves along time \(\zeta(t)\).Path integral calculation of heat kernel traces with first order operator insertionshttps://zbmath.org/1472.811332021-11-25T18:46:10.358925Z"Bastianelli, Fiorenzo"https://zbmath.org/authors/?q=ai:bastianelli.fiorenzo"Comberiati, Francesco"https://zbmath.org/authors/?q=ai:comberiati.francescoSummary: We study generalized heat kernel coefficients, which appear in the trace of the heat kernel with an insertion of a first-order differential operator, by using a path integral representation. These coefficients may be used to study gravitational anomalies, i.e. anomalies in the conservation of the stress tensor. We use the path integral method to compute the coefficients related to the gravitational anomalies of theories in a non-abelian gauge background and flat space of dimensions 2, 4, and 6. In 4 dimensions one does not expect to have genuine gravitational anomalies. However, they may be induced at intermediate stages by regularization schemes that fail to preserve the corresponding symmetry. A case of interest has recently appeared in the study of the trace anomalies of Weyl fermions.A unified construction of Skyrme-type non-linear sigma models via the higher dimensional Landau modelshttps://zbmath.org/1472.811452021-11-25T18:46:10.358925Z"Hasebe, Kazuki"https://zbmath.org/authors/?q=ai:hasebe.kazukiSummary: A curious correspondence has been known between Landau models and non-linear sigma models: Reinterpreting the base-manifolds of Landau models as field manifolds, the Landau models are transformed to non-linear sigma models with same global and local symmetries. With the idea of the dimensional hierarchy of higher dimensional Landau models, we exploit this correspondence to present a systematic procedure for construction of non-linear sigma models in higher dimensions. We explicitly derive \(O(2k+1)\) non-linear sigma models in \(2k\) dimension based on the parent tensor gauge theories that originate from non-Abelian monopoles. The obtained non-linear sigma models turn out to be Skyrme-type non-linear sigma models with \(O(2k)\) local symmetry. Through a dimensional reduction of Chern-Simons tensor field theories, we also derive Skyrme-type \(O(2k)\) non-linear sigma models in \(2 k - 1\) dimension, which realize the original and other Skyrme models as their special cases. As a unified description, we explore Skyrme-type \(O(d+1)\) non-linear sigma models and clarify their basic properties, such as stability of soliton configurations, scale invariant solutions, and field configurations with higher winding number.Universal effective couplings of the three-dimensional \(n\)-vector model and field theoryhttps://zbmath.org/1472.811582021-11-25T18:46:10.358925Z"Kudlis, A."https://zbmath.org/authors/?q=ai:kudlis.andrey"Sokolov, A. I."https://zbmath.org/authors/?q=ai:sokolov.aleksandr-iSummary: We calculate the universal ratios \(R_{2k}\) of renormalized coupling constants \(g_{2k}\) entering the critical equation of state for the generalized Heisenberg (three-dimensional \(n\)-vector) model. Renormalization group (RG) expansions of \(R_8\) and \(R_{10}\) for arbitrary \(n\) are found in the four-loop and three-loop approximations respectively. Universal octic coupling \(R_8^\ast\) is estimated for physical values of spin dimensionality \(n = 0, 1, 2, 3\) and for \(n = 4, \dots 64\) to get an idea about asymptotic behavior of \(R_8^\ast \). Its numerical values are obtained by means of the resummation of the RG series and within the pseudo-\(\varepsilon\) expansion approach. Regarding \(R_{10}\) our calculations show that three-loop RG and pseudo-\(\varepsilon\) expansions possess big and rapidly growing coefficients for physical values of \(n\) what prevents getting fair numerical estimates.Stability in the higher derivative abelian gauge field theorieshttps://zbmath.org/1472.811642021-11-25T18:46:10.358925Z"Dai, Jialiang"https://zbmath.org/authors/?q=ai:dai.jialiangSummary: We present an exact derivation of conserved tensors associated to the higher-order symmetries in the higher derivative Abelian gauge field theories. In our model, the wave operator of the derived theory is a \(n\)-th order polynomial expressed in terms of the usual Maxwell operator. Relying on this formalism and utilizing the extension of Noether's theorem, we acquire a series of conserved second-rank tensors which includes the standard canonical energy-momentum tensors. Moreover, with the aid of auxiliary fields, we succeed in obtaining the relations between the root decomposition of characteristic polynomial of the wave operator and the conserved energy-momentum tensors in the context of another equivalent lower-order representation. Under the certain conditions, although the canonical energy of the higher derivative dynamics is unbounded from below, the 00-component of the linear combination of these conserved quantities is bounded. By this reason, the original derived theory is considered stable. Finally, as an instructive example, we elaborate the third-order derived system and analyze the stabilities in different cases of root decomposition of the characteristic polynomial extensively.Holographic unitary renormalization group for correlated electrons. II: Insights on fermionic criticalityhttps://zbmath.org/1472.811762021-11-25T18:46:10.358925Z"Mukherjee, Anirban"https://zbmath.org/authors/?q=ai:mukherjee.anirban"Lal, Siddhartha"https://zbmath.org/authors/?q=ai:lal.siddharthaSummary: Capturing the interplay between electronic correlations and many-particle entanglement requires a unified framework for Hamiltonian and eigenbasis renormalization. In this work, we apply the unitary renormalization group (URG) scheme developed in a companion work [the authors, ibid. 960, Article ID 115170, 72 p. (2020; Zbl 1472.81177)] to the study of two archetypal models of strongly correlated lattice electrons, one with translation invariance and one without. We obtain detailed insight into the emergence of various gapless and gapped phases of quantum electronic matter by computing effective Hamiltonians from numerical evaluation of the various RG equations, as well as their entanglement signatures through their respective tensor network descriptions. For the translationally invariant model of a single-band of interacting electrons, this includes results on gapless metallic phases such as the Fermi liquid and Marginal Fermi liquid, as well as gapped phases such as the reduced Bardeen-Cooper-Schrieffer, pair density-wave and Mott liquid phases. Additionally, a study of a generalised Sachdev-Ye model with disordered four-fermion interactions offers detailed results on many-body localised phases, as well as thermalised phase. We emphasise the distinctions between the various phases based on a combined analysis of their dynamical (obtained from the effective Hamiltonian) and entanglement properties. Importantly, the RG flow of the Hamiltonian vertex tensor network is shown to lead to emergent gauge theories for the gapped phases. Taken together with results on the holographic spacetime generated from the RG of the many-particle eigenstate (seen through, for instance, the holographic upper bound of the one-particle entanglement entropy), our analysis offer an ab-initio perspective of the gauge-gravity duality for quantum liquids that are emergent in systems of correlated electrons.Holographic unitary renormalization group for correlated electrons. I: A tensor network approachhttps://zbmath.org/1472.811772021-11-25T18:46:10.358925Z"Mukherjee, Anirban"https://zbmath.org/authors/?q=ai:mukherjee.anirban"Lal, Siddhartha"https://zbmath.org/authors/?q=ai:lal.siddharthaSummary: We present a unified framework for the renormalisation of the Hamiltonian and eigenbasis of a system of correlated electrons, unveiling thereby the interplay between electronic correlations and many-particle entanglement. For this, we extend substantially the unitary renormalization group (URG) scheme introduced in [the authors, ``Scaling theory for Mott-Hubbard transitions: I. \(T = 0\) phase diagram of the 1/2-filled Hubbard model'', New J. Phys. 22, No. 6, Article ID 063007, 26 p. (2020; \url{doi:10.1088/1367-2630/ab8831}); ``Scaling theory for Mott-Hubbard transitions. II: Quantum criticality of the doped Mott insulator'', ibid. 22, No. 6, Article ID 063008, 25 p. (2020; \url{doi:10.1088/1367-2630/ab890c}); ``Holographic entanglement renormalisation of topological order in a quantum liquid'', Preprint, \url{arXiv:2003.06118}]. We recast the RG as a discrete flow of the Hamiltonian tensor network, i.e., the collection of various \(2n\)-point scattering vertex tensors comprising the Hamiltonian. The renormalisation progresses via unitary transformations that block diagonalizes the Hamiltonian iteratively via the disentanglement of single-particle eigenstates. This procedure incorporates naturally the role of quantum fluctuations. The RG flow equations possess a non-trivial structure, displaying a feedback mechanism through frequency-dependent dynamical self-energies and correlation energies. The interplay between various UV energy scales enables the coupled RG equations to flow towards a stable fixed point in the IR. The effective Hamiltonian at the IR fixed point generically has a reduced parameter space, as well as number of degrees of freedom, compared to the microscopic Hamiltonian. Importantly, the vertex RG flows are observed to govern the RG flow of the tensor network that denotes the coefficients of the many-particle eigenstates. The RG evolution of various many-particle entanglement features of the eigenbasis are, in turn, quantified through the coefficient tensor network. In this way, we show that the URG framework provides a microscopic understanding of holographic renormalisation: the RG flow of the vertex tensor network generates a eigenstate coefficient tensor network possessing a many-particle entanglement metric. We find that the eigenstate tensor network accommodates sign factors arising from fermion exchanges, and that the IR fixed point reached generically involves a trivialisation of the fermion sign factor. Several results are presented for the emergence of composite excitations in the neighbourhood of a gapless Fermi surface, as well as for the condensation phenomenon involving the gapping of the Fermi surface.
For Part II, see the authors [Nucl. Phys., B 960, Article ID 115163, 63 p. (2020; Zbl 1472.81176)].\(O(d,d)\) transformations preserve classical integrabilityhttps://zbmath.org/1472.812272021-11-25T18:46:10.358925Z"Orlando, Domenico"https://zbmath.org/authors/?q=ai:orlando.domenico"Reffert, Susanne"https://zbmath.org/authors/?q=ai:reffert.susanne"Sekiguchi, Yuta"https://zbmath.org/authors/?q=ai:sekiguchi.yuta"Yoshida, Kentaroh"https://zbmath.org/authors/?q=ai:yoshida.kentarohSummary: In this note, we study the action of \(O(d, d)\) transformations on the integrable structure of two-dimensional non-linear sigma models via the doubled formalism. We construct the Lax pairs associated with the \(O(d, d)\)-transformed model and find that they are in general non-local because they depend on the winding modes. We conclude that every \(O(d, d; \mathbb{R})\) deformation preserves integrability. As an application we compute the Lax pairs for continuous families of deformations, such as \(J \bar{J}\) marginal deformations and TsT transformations of the three-sphere with \(H\)-flux.Time-space noncommutativity and Casimir effecthttps://zbmath.org/1472.812332021-11-25T18:46:10.358925Z"Harikumar, E."https://zbmath.org/authors/?q=ai:harikumar.e"Panja, Suman Kumar"https://zbmath.org/authors/?q=ai:panja.suman-kumar"Rajagopal, Vishnu"https://zbmath.org/authors/?q=ai:rajagopal.vishnuSummary: We show that the Casimir force and energy are modified in the \(\kappa\)-deformed space-time. This is shown by solving the Green's function corresponding to \(\kappa \)-deformed scalar field equation in presence of two parallel plates, modelled by \(\delta\)-function potentials. Exploiting the relation between Energy-Momentum tensor and Green's function, we calculate corrections to Casimir force, valid up to second order in the deformation parameter. The Casimir force is shown to get corrections which scale as \(L^{- 4}\) and \(L^{- 6}\) and both these types of corrections produce attractive forces. Using the measured value of Casimir force, we show that the deformation parameter should be below \(10^{-23}\) m.Scattering, spectrum and resonance states completeness for a quantum graph with Rashba Hamiltonianhttps://zbmath.org/1472.812512021-11-25T18:46:10.358925Z"Blinova, Irina V."https://zbmath.org/authors/?q=ai:blinova.irina-v"Popov, Igor Y."https://zbmath.org/authors/?q=ai:popov.igor-yu"Smolkina, Maria O."https://zbmath.org/authors/?q=ai:smolkina.maria-oSummary: Quantum graphs consisting of a ring with two semi-infinite edges attached to the same point of the ring is considered. We deal with the Rashba spin-orbit Hamiltonian on the graph. A theorem concerning to completeness of the resonance states on the ring is proved. Due to use of a functional model, the problem reduces to factorization of the characteristic matrix-function. The result is compared with the corresponding completeness theorem for the Schrödinger, Dirac and Landau quantum graphs.
For the entire collection see [Zbl 1471.47002].Covariant kinetic theory and transport coefficients for Gribov plasmahttps://zbmath.org/1472.812722021-11-25T18:46:10.358925Z"Jaiswal, Amaresh"https://zbmath.org/authors/?q=ai:jaiswal.amaresh"Haque, Najmul"https://zbmath.org/authors/?q=ai:haque.najmulSummary: Gribov quantization is a method to improve the infrared dynamics of Yang-Mills theory. We study the thermodynamics and transport properties of a plasma consisting of gluons whose propagator is improved by the Gribov prescription. We first construct thermodynamics of Gribov plasma using the gauge invariant Gribov dispersion relation for interacting gluons. When the Gribov parameter in the dispersion relation is temperature dependent, one expects a mean field correction to the Boltzmann equation. We formulate covariant kinetic theory for the Gribov plasma and determine the mean-field contribution in the Boltzmann equation. This leads to a quasiparticle like framework with a bag correction to pressure and energy density which mimics confinement. The temperature dependence of the Gribov parameter and bag pressure is fixed by matching with lattice results for a system of gluons. Finally we calculate the temperature dependence of the transport coefficients, i.e., bulk and shear viscosities.Solutions to the minimization problem arising in a dark monopole model in gauge field theoryhttps://zbmath.org/1472.813042021-11-25T18:46:10.358925Z"Zhang, Xiangqin"https://zbmath.org/authors/?q=ai:zhang.xiangqin"Yang, Yisong"https://zbmath.org/authors/?q=ai:yang.yisongSummary: We prove the existence of dark monopole solutions in a recently formulated Yang-Mills-Higgs theory model with technical features similar to the classical monopole problems. The solutions are obtained as energy-minimizing static spherically symmetric field configurations of unit topological charge. We overcome the difficulty of recovering the full set of boundary conditions by a regularization method which may be applied to other more complicated problems concerning monopoles and dyons in non-Abelian gauge field theories. Furthermore we show in a critical coupling situation that an explicit BPS solution may be used to provide energy estimates for non-BPS monopole solutions. Besides, in the limit of infinite Higgs coupling parameter, although no explicit construction is available, we establish an existence and uniqueness result for a monopole solution and obtain its energy bounds.Fundamental properties of the proton in light-front zero modeshttps://zbmath.org/1472.813122021-11-25T18:46:10.358925Z"Ji, Xiangdong"https://zbmath.org/authors/?q=ai:ji.xiangdongSummary: For a proton in the infinite momentum frame, its wave function contains a zero-momentum part (light-front zero-modes) originated from the modification of the QCD vacuum in the presence of the valence quarks, exhibiting a light-front long-range order in the quantum state. This ``non-travelling'' component of the proton contributes to its fundamental properties, including the mass and spin, as well as the naive-time-reversal-odd correlations between transverse momentum and spin. The large momentum effective theory (LaMET) provides a theoretical approach to study the physical properties of the proton's zero-mode ``condensate'' through lattice simulations.Some properties of the potential-to-ground state map in quantum mechanicshttps://zbmath.org/1472.813222021-11-25T18:46:10.358925Z"Garrigue, Louis"https://zbmath.org/authors/?q=ai:garrigue.louisThe author considers properties of the map from potential to the ground state in many-body quantum mechanics. External potentials \(v\in L^p + L^\infty\) and interaction potentials \(w \in L^p + L^\infty\) for \(p> \max(2d/3, 2)\) where \(d\) is the dimension of the underlying space are considered. The first result is that the space of binding potentials is path-connected. Then the author shows that the map from potentials to the ground state is locally weak-strong continuous and that its differential is compact. This implies that the Kohn-Sham inverse problem in Density Functional Theory is ill-posed on a bounded set.Many-body work distributionshttps://zbmath.org/1472.813232021-11-25T18:46:10.358925Z"Kheirandish, Fardin"https://zbmath.org/authors/?q=ai:kheirandish.fardinSummary: The work distribution function for a non-relativistic, non-interacting quantum many-body system interacting with classical external sources is investigated. Exact expressions for the characteristic function corresponding to the work distribution function is obtained for arbitrary switching function and coupling functions. The many-body frequencies are assumed to be generally time-dependent in order to take into account the possibility of moving the boundaries of the system in a predefined process linking the characteristic function to the fluctuation-induced energies in confined geometries. Some limiting cases are considered and discussed.The intrinsical character of the electronic correlation in an electron gashttps://zbmath.org/1472.813242021-11-25T18:46:10.358925Z"Liu, Yu-Liang"https://zbmath.org/authors/?q=ai:liu.yuliangSummary: By introducing the phase transformation of electron operators, we map the equation of motion of an one-particle Green's function into that of a non-interacting one-particle Green's function where the electrons are moving in a time-depending scalar potential and pure gauge fields for a D-dimensional electron gas, and we demonstrate that the electronic correlation strength strongly depends upon the excitation energy spectrum and collective excitation modes of electrons. It naturally explains that the electronic correlation strength is strong in the one dimension, while it is weak in the three dimensions.On the effect of fractional statistics on quantum ion acoustic waveshttps://zbmath.org/1472.813272021-11-25T18:46:10.358925Z"Ourabah, Kamel"https://zbmath.org/authors/?q=ai:ourabah.kamelSummary: In this paper, I study the effect of a small deviation from the Fermi-Dirac statistics on the quantum ion acoustic waves. For this purpose, a quantum hydrodynamic model is developed based on the Polychronakos statistics, which allows for a smooth interpolation between the Fermi and Bose limits, passing through the case of classical particles. The model includes the effect of pressure as well as quantum diffraction effects through the Bohm potential. The equation of state for electrons obeying fractional statistics is obtained and the effect of fractional statistics on the kinetic energy and the coupling parameter is analyzed. Through the model, the effect of fractional statistics on the quantum ion acoustic waves is highlighted, exploring both linear and weakly nonlinear regimes. It is found that fractional statistics enhance the amplitude and diminish the width of the quantum ion acoustic waves. Furthermore, it is shown that a small deviation from the Fermi-Dirac statistics can modify the type structures, from bright to dark soliton. All known results of fully degenerate and non-degenerate cases are reproduced in the proper limits.Soliton lattices in the Gross-Pitaevskii equation with nonlocal and repulsive couplinghttps://zbmath.org/1472.813322021-11-25T18:46:10.358925Z"Sakaguchi, Hidetsugu"https://zbmath.org/authors/?q=ai:sakaguchi.hidetsuguSummary: Spatially-periodic patterns are studied in nonlocally coupled Gross-Pitaevskii equation. We show first that spatially periodic patterns appear in a model with the dipole-dipole interaction. Next, we study a model with a finite-range coupling, and show that a repulsively coupled system is closely related with an attractively coupled system and its soliton solution becomes a building block of the spatially-periodic structure. That is, the spatially-periodic structure can be interpreted as a soliton lattice. An approximate form of the soliton is given by a variational method. Furthermore, the effects of the rotating harmonic potential and spin-orbit coupling are numerically studied.The linear dynamics of wave functions in causal fermion systemshttps://zbmath.org/1472.813342021-11-25T18:46:10.358925Z"Finster, Felix"https://zbmath.org/authors/?q=ai:finster.felix"Kamran, Niky"https://zbmath.org/authors/?q=ai:kamran.niky"Oppio, Marco"https://zbmath.org/authors/?q=ai:oppio.marcoThe goal of the paper is to study the dynamics of the physical wave functions of a causal fermion system. Due to nonlinearity of the causal action all the wave functions interact with each other. The authors interpretes this in a manner similar to back reaction of a quantum spinor field in a curved spacetime on the metric. The authors derive dynamical wave function. They show that its solution form a Hilbert space, whose salar product is represented by a conserved layer integral. By few theorems the authors prove that the initial value problem for the dynamical wave equation admits a unique global solution. Subsequently they construct causal Green' s operators and investigate their properties. As a particular example the authors investigate the regularized Minkowski vacuum. The article includes 6 sections from which one has a preliminary character. Two apendices are included in order to clarify the role of comutator jets to which a previous section was concerned.Comparing light-front quantization with instant-time quantizationhttps://zbmath.org/1472.813382021-11-25T18:46:10.358925Z"Mannheim, Philip D."https://zbmath.org/authors/?q=ai:mannheim.philip-d"Lowdon, Peter"https://zbmath.org/authors/?q=ai:lowdon.peter"Brodsky, Stanley J."https://zbmath.org/authors/?q=ai:brodsky.stanley-jSummary: In this paper we compare light-front quantization and instant-time quantization both at the level of operators and at the level of their Feynman diagram matrix elements. At the level of operators light-front quantization and instant-time quantization lead to equal light-front time commutation (or anticommutation) relations that appear to be quite different from equal instant-time commutation (or anticommutation) relations. Despite this we show that at unequal times instant-time and light-front commutation (or anticommutation) relations actually can be transformed into each other, with it only being the restriction to equal times that makes the commutation (or anticommutation) relations appear to be so different. While our results are valid for both bosons and fermions, for fermions there are subtleties associated with tip of the light cone contributions that need to be taken care of. At the level of Feynman diagrams we show for non-vacuum Feynman diagrams that the pole terms in four-dimensional light-front Feynman diagrams reproduce the widely used three-dimensional light-front on-shell Hamiltonian Fock space formulation in which the light-front energy and light-front momentum are on shell. Moreover, we show that the contributions of pole terms in non-vacuum instant-time and non-vacuum light-front Feynman diagrams are equal. However, because of circle at infinity contributions we show that this equivalence of pole terms fails for four-dimensional light-front vacuum tadpole diagrams. Then, and precisely because of these circle at infinity contributions, we show that light-front vacuum tadpole diagrams are not only nonzero, they quite remarkably are actually equal to the pure pole term instant-time vacuum tadpole diagrams. Light-front vacuum diagrams are not correctly describable by the on-shell Hamiltonian formalism, and thus not by the closely related infinite momentum frame prescription either. Thus for the light-front vacuum sector we must use the off-shell Feynman formalism as it contains information that is not accessible in the on-shell Hamiltonian Fock space approach. We show that light-front quantization is intrinsically nonlocal, and that for fermions this nonlocality is present in Ward identities. One can project fermion spinors into so-called good and bad components, and both of these components contribute in Ward identities. Central to our analysis is that the transformation from instant-time coordinates and fields to light-front coordinates and fields is a unitary, spacetime-dependent translation. Consequently, not only are instant-time quantization and light-front quantization equivalent, because of general coordinate invariance they are unitarily equivalent.On the boundary layer equations with phase transition in the kinetic theory of gaseshttps://zbmath.org/1472.820112021-11-25T18:46:10.358925Z"Bernhoff, Niclas"https://zbmath.org/authors/?q=ai:bernhoff.niclas"Golse, François"https://zbmath.org/authors/?q=ai:golse.francoisIn the present paper, the authors deal with the nonlinear half-space problem for the Boltzmann equation written in terms of the relative fluctuation of distribution function about the normalized Maxwellian \(M\) \[ \left\{ \begin{array}{cc} (\xi _i+u)\partial _xf_u+\mathcal{L}f_u, & \xi \in \mathbb{R}^3,x>0, \\
f_u(0,\xi )=f_b(\xi ), & \xi _i+u>0. \end{array} \right. \] The authors prove the existence of the curve \(C\) corresponding to solutions of the equations given in some neighborhood of the point \((1, 0, 1)\) converging as \(x\rightarrow \infty\) with exponential speed uniformly in \(u\). The authors provides a self-contained construction of the solution to the Nicolaenko-Thurber generalized eigenvalue problem near \(u = 0\). Then, the authors introduce the penalization method, and formulate the problem to be solved by a fixed point argument. The linearized penalized problem is studied. Also, the authors investigate the (weakly) nonlinear penalized problem by a fixed point argument. The authors give an alternative, possibly simpler proof of one of the results discussed in [\textit{T.-P. Liu} and \textit{S.-H. Yu}, Arch. Ration. Mech. Anal. 209, No. 3, 869--997 (2013; Zbl 1290.35181)].Spectral continuity for aperiodic quantum systems: applications of a folklore theoremhttps://zbmath.org/1472.820152021-11-25T18:46:10.358925Z"Beckus, Siegfried"https://zbmath.org/authors/?q=ai:beckus.siegfried"Bellissard, Jean"https://zbmath.org/authors/?q=ai:bellissard.jean-v"De Nittis, Giuseppe"https://zbmath.org/authors/?q=ai:de-nittis.giuseppeThe authors provide a necessary and sufficient condition for a subshift to admit periodic approximations in the Hausdorff topology. Moreover, a rigorous justification for the accuracy and reliability of algorithmic methods that are used to numerically compute the spectra of certain self-adjoint operators, namely Hamiltonians associated with subshifts that admit periodic approximations, is given.Generation of discrete structures in phase-space via charged particle trapping by an electrostatic wavehttps://zbmath.org/1472.820252021-11-25T18:46:10.358925Z"Vainchtein, Dmitri"https://zbmath.org/authors/?q=ai:vainchtein.dmitri-l"Fridman, Greg"https://zbmath.org/authors/?q=ai:fridman.greg"Artemyev, Anton"https://zbmath.org/authors/?q=ai:artemyev.anton-vSummary: The wave-particle resonant interaction plays an important role in the charged particle energization by trapping (capture) into resonance. For the systems with waves propagating through inhomogeneous plasma, the key small parameter is the ratio of the wave wavelength to a characteristic spatial scale of inhomogeneity. When that parameter is very small, the asymptotic methods are applicable for the system description, and the resultant energy distribution of trapped particle ensemble has a typical Gaussian profile around some mean value. However, for moderate values of that parameter, the energy distribution has a fine structure including several maxima, each corresponding to the discrete number of oscillations a particle makes in the trapped state. We explain this novel effect which can play important role for generation of unstable distributions of accelerated particles in many space plasma systems.A neural network-based policy iteration algorithm with global \(H^2\)-superlinear convergence for stochastic games on domainshttps://zbmath.org/1472.820302021-11-25T18:46:10.358925Z"Ito, Kazufumi"https://zbmath.org/authors/?q=ai:ito.kazufumi"Reisinger, Christoph"https://zbmath.org/authors/?q=ai:reisinger.christoph"Zhang, Yufei"https://zbmath.org/authors/?q=ai:zhang.yufeiThe following Hamilton-Jacobi-Bellman-Isaacs (HJBI) nonhomogeneous Dirichlet boundary value problem is considered: $F(u): =-a^{ij}(x)\partial_{ij}u+ G(x,u,\nabla u)=0$, for a.e. $x\in \Omega$, $\tau u=g$, on $\partial\Omega$, with a nonlinear Hamiltonian, $G(x,u,\nabla u)=\max_{\alpha \in A}\min_{\beta \in B}(b^i(x,\alpha,\beta)$ $\partial_iu(x)+c(x,\alpha,\beta)u(x) -f(x,\alpha,\beta))$. The aim here is to investigate some numerical algorithms for solving this kind of problems. The second section is devoted to basics. Under some assumptions on the coefficients, the uniqueness of the strong solution in $H^2(\Omega)$ is proved. In the third section one presents the policy iteration algorithm -- Algorithm 1 -- for the Dirichlet problem, followed by the convergence analysis. Results on semi smoothness of the HJBI operator, q-superlinear convergence of Algorithm 1 and global convergence of Algorithm 1 are proved. In the fourth section the authors develop an inexact policy algorithm for the stated Dirichlet problem. The idea is to compute an approximate solution for the linear Dirichlet problem for the iteration $u^{k+1}\in H^2(\Omega)$ in Algorithm 1, by solving an optimization problem over a set of trial functions, within a given accuracy. The new inexact policy iteration algorithm for the Dirichlet problem -- Algorithm 2 -- is presented and under some special assumptions a result on global superlinear convergence is proved. In the fifth section we find an extension of the developed iteration scheme to other boundary value problems and a connection to the artificial neural network technology. One considers a HJBI oblique derivative problem
\[
F(u): =-a^{ij}(x)\partial _{ij}u+G(x,u,\nabla u)=0,\text{ for a.e. }x\in\Omega,
\]
$Bu:=\gamma^i\tau(\partial_iu)+\gamma^0$ $\tau u-g$, on $\partial\Gamma$. Under some assumptions on the coefficients, one proves that the oblique derivative problem admits a unique strong solution in $H^2(\Omega)$. For solving the oblique derivative problem one develops a neural network-based policy iteration algorithm, Algorithm 3. The global superlinear convergence of Algorithm 3 is proved. In the sixth section, there is a large discussion on applications of the developed algorithms to the stochastic Zermelo navigation problem. Some fundamental results used in the article are resumed at the end of the paper.Gramian solutions and soliton interactions for a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in a plasma or fluidhttps://zbmath.org/1472.820352021-11-25T18:46:10.358925Z"Chen, Su-Su"https://zbmath.org/authors/?q=ai:chen.su-su"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.boSummary: Plasmas and fluids are of current interest, supporting a variety of wave phenomena. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation for the nonlinear phenomena in a plasma or fluid. Based on the existing bilinear form, \(N\)-soliton solutions in the Gramian are derived, where \(N = 1, 2, 3\)\dots{}. With \(N = 3\), three-soliton solutions are constructed. Fission and fusion for the three solitons are presented. Effects of the variable coefficients, i.e. \(h(t), l(t), q(t), n(t)\) and \(m(t)\), on the soliton fission and fusion are revealed: soliton velocity is related to \(h(t), l(t), q(t), n(t)\) and \(m(t)\), while the soliton amplitude cannot be affected by them, where \(t\) is the scaled temporal coordinate, \(h(t), l(t)\) and \(q(t)\) give the perturbed effects, and \(m(t)\) and \(n(t)\), respectively, stand for the disturbed wave velocities along two transverse spatial coordinates. We show the three parallel solitons with the same direction.On the Nernst-Planck-Navier-Stokes systemhttps://zbmath.org/1472.820372021-11-25T18:46:10.358925Z"Constantin, Peter"https://zbmath.org/authors/?q=ai:constantin.peter"Ignatova, Mihaela"https://zbmath.org/authors/?q=ai:ignatova.mihaelaIn this paper the authors study and obtain results about global existence and stability properties and convergence properties to Boltzmann states of a model for ionic electrodiffusion in fluids, in bounded domains for various boundary conditions and a wide class of initial data. For the Nernst-Planck-Navier-Stokes model the Navier-Stokes and Poisson equations are considered. This extends earlier known results for smaller classes of initial data and local existence. The model has various applications in the physics literature.Mass-based finite volume scheme for aggregation, growth and nucleation population balance equationhttps://zbmath.org/1472.820442021-11-25T18:46:10.358925Z"Singh, Mehakpreet"https://zbmath.org/authors/?q=ai:singh.mehakpreet"Ismail, Hamza Y."https://zbmath.org/authors/?q=ai:ismail.hamza-y"Matsoukas, Themis"https://zbmath.org/authors/?q=ai:matsoukas.themis"Albadarin, Ahmad B."https://zbmath.org/authors/?q=ai:albadarin.ahmad-b"Walker, Gavin"https://zbmath.org/authors/?q=ai:walker.gavinSummary: In this paper, a new mass-based numerical method is developed using the notion of \textit{L. Forestier-Coste} and \textit{S. Mancini} [SIAM J. Sci. Comput. 34, No. 6, B840--B860 (2012; Zbl 1259.82054)] for solving a one-dimensional aggregation population balance equation. The existing scheme requires a large number of grids to predict both moments and number density function accurately, making it computationally very expensive. Therefore, a mass-based finite volume is developed which leads to the accurate prediction of different integral properties of number distribution functions using fewer grids. The new mass-based and existing finite volume schemes are extended to solve simultaneous aggregation-growth and aggregation-nucleation problems. To check the accuracy and efficiency, the mass-based formulation is compared with the existing method for two kinds of benchmark kernels, namely analytically solvable and practical oriented kernels. The comparison reveals that the mass-based method computes both number distribution functions and moments more accurately and efficiently than the existing method.Curvature of space and time, with an introduction to geometric analysishttps://zbmath.org/1472.830012021-11-25T18:46:10.358925Z"Stavrov, Iva"https://zbmath.org/authors/?q=ai:stavrov.ivaThe book grew out of a summer program at the Park City Math institute (PCMI) in 2013 and a class at Lewis and Clark college in 2015. It was the aim of the courses to introduce students to general relativity and geometric analysis at an undergraduate level.
The book follows an unusual track, concepts from Riemannian geometry are introduced without refering to results from differential topology. Hence the standard definitions of manifolds and tangent vectors are not presented. This is not by accident but follows the belief of the author stated in the preface that \textit{the practice of organizing information in a deductive manner, which as mathematicians we are committed to, is not always conducive to learning mathematics and developing intuition.}
The book is divided into five chapters, the topic of the first chapter is \textit{Introduction to Riemannian geometry.} It starts with a quotation from Riemann's habilitation lecture presented in 1854 in Göttingen. The concept of a Riemannian metric is introduced by discussing the examples of the euclidean metric, the round metric on a sphere and the standard metric of the real projective plane. The concept of a manifold is introduced by using different coordinate systems with certain transition maps. Geodesics are introduced using their variational characterization. The geodesic equation then leads to the introduction of Christoffel symbols and geodesically completness is discussed shortly.
The title of chapter 2 is \textit{Differential calculus with tensors.} Here directional derivatives of functions, vector fields and the Levi-Civita connection are discussed and motivated. Then tensor fields and their coordinate expression are presented. The gradient of a function, the divergence of a vector field and the Laplacian of a function are defined, Green's identities as well as the maximum principle are formulated. The next section deals with the differentiation of tensors and motivates the definition of the curvature tensor. Then the symmetries of the curvature tensor and the Bianchi identities are given.
The topic of the third chapter is \textit{Curvature.} Here the connection with Jacobi fields as variational vector fields of variations by geodesics is emphasized. Ricci curvature and scalar curvature are introduced and their geometric meaning is explained.
The fourth chapter is named \textit{General relativity.} Concepts from special relativity are presented in the first section, whereas the next section deals with \textit{Gravity and general relativity}. It motivates the Einstein equation and also presents its initial value formulation. In the next two sections the geometry of the Schwarzschild spacetime and its Kruskal Szekeres extension is discussed. In this chapter the relativistic Poisson equation occurs.
In the last chapter titled \textit{Introduction to geometric analysis} ideas of the proof of the existence of a solution of the relativistic Poisson equation are given. This is meant to be an introduction to geometric analysis. In the final section the concept of the ADM-mass of an asymptotically Euclidean metric is motivated.
Without doubt it is a remarkable book. It deals with an enormous amount of mathematical contents, it provides excellent motivations and insights starting from typical examples. Often proofs are not presented, but there are a lot of excercises following each section with many useful explainations. The book gives an impressing overview on Riemannian geometry, general relativity and geometric analysis. It supplements standard books or courses on the topics of the book. And it may invite readers to study parts of the material in detail. It is likely that readers agree with the author's statement from the preface: \textit{An approach to teaching and learning mathematics which relies on several passes through a subject, each at a higher level of mathematical rigor, is in a sense historically proven to be pedagogically optimal.}Static spherically symmetric Einstein-Vlasov bifurcations of the Schwarzschild spacetimehttps://zbmath.org/1472.830222021-11-25T18:46:10.358925Z"Jabiri, Fatima Ezzahra"https://zbmath.org/authors/?q=ai:jabiri.fatima-ezzahraA static spherically symmetric solutions to the Einstein-Vlasov system containing a matter shell located in the exterior region of the Schwarzschild black hole is considered. The proof is based on the study of trapped timelike geodesics of a perturbed Schwarzschild spacetime. Basic back-ground material on the Einstein-Vlasov system and the geodesic motion in the Schwarzschild exterior is presented.Variational analysis of landscape elevation and drainage networkshttps://zbmath.org/1472.860052021-11-25T18:46:10.358925Z"Hooshyar, Milad"https://zbmath.org/authors/?q=ai:hooshyar.milad"Anand, Shashank"https://zbmath.org/authors/?q=ai:anand.shashank"Porporato, Amilcare"https://zbmath.org/authors/?q=ai:porporato.amilcareSummary: Landscapes evolve towards surfaces with complex networks of channels and ridges in response to climatic and tectonic forcing. Here, we analyse variational principles giving rise to minimalist models of landscape evolution as a system of partial differential equations that capture the essential dynamics of sediment and water balances. Our results show that in the absence of diffusive soil transport the steady-state surface extremizes the average domain elevation. Depending on the exponent \(m\) of the specific drainage area in the erosion term, the critical surfaces are either minima \((0 < m < 1)\) or maxima \((m > 1)\), with \(m = 1\) corresponding to a saddle point. We establish a connection between landscape evolution models and optimal channel networks and elucidate the role of diffusion in the governing variational principles.Analytic resolution of time-domain half-space Green's functions for internal loads by a displacement potential-Laplace-Hankel-cagniard transform methodhttps://zbmath.org/1472.860102021-11-25T18:46:10.358925Z"Pak, Ronald Y. S."https://zbmath.org/authors/?q=ai:pak.ronald-y-s"Bai, Xiaoyong"https://zbmath.org/authors/?q=ai:bai.xiaoyongSummary: A refined yet compact analytical formulation is presented for the time-domain elastodynamic response of a three-dimensional half-space subject to an arbitrary internal or surface force distribution. By integrating Laplace and Hankel transforms into a method of displacement potentials and Cagniard's inversion concept, it is shown that the solution can be derived in a straightforward manner for the generalized classical wave propagation problem. For the canonical case of a buried point load with a step time function, the response is proved to be naturally reducible with the aid of a parametrized Bessel function integral representation to six wave-group integrals on finite contours in the complex plane that stay away from all branch points and the Rayleigh pole except possibly at the starting point of the contours. On the latter occasions, the possible singularities of the integrals can be rigorously extracted by an extended method of asymptotic decomposition, rendering the residual numerical computation a simple exercise. With the new solution format, the arrival time of each wave group is derivable by simple criteria on the contour. Typical results for the time-domain response for an internal point force as well as the degenerate case of a surface point source are included for comparison and illustrations.Ice sheet flow with thermally activated sliding. I: The role of advectionhttps://zbmath.org/1472.860382021-11-25T18:46:10.358925Z"Mantelli, E."https://zbmath.org/authors/?q=ai:mantelli.elisa"Haseloff, M."https://zbmath.org/authors/?q=ai:haseloff.m"Schoof, C."https://zbmath.org/authors/?q=ai:schoof.christianSummary: Flow organization into systems of fast-moving ice streams is a well-known feature of ice sheets. Fast motion is frequently the result of sliding at the base of the ice sheet. Here, we consider how this basal sliding is first initiated as the result of changes in bed temperature. We show that an abrupt sliding onset at the melting point, with no sliding possible below that temperature, leads to rapid drawdown of cold ice and refreezing as the result of the increased temperature gradient within the ice, and demonstrate that this result holds regardless of the mechanical model used to describe the flow of ice. Using this as a motivation, we then consider the possibility of a region of `subtemperate sliding' in which sliding at reduced velocities occurs in a narrow range of temperatures just below the melting point. We confirm that this prevents the rapid drawdown of ice and refreezing of the bed, and construct a simple numerical method for computing steady-state ice sheet profiles that include a subtemperate region. The stability of such an ice sheet is analysed in a companion paper.
The Part II, see [the first and the third author, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 475, No. 2231, Article ID 20190411, 25 p. (2019; Zbl 1472.86039)].Ice sheet flow with thermally activated sliding. II: The stability of subtemperate regionshttps://zbmath.org/1472.860392021-11-25T18:46:10.358925Z"Mantelli, E."https://zbmath.org/authors/?q=ai:mantelli.elisa"Schoof, C."https://zbmath.org/authors/?q=ai:schoof.christianSummary: The onset of sliding in ice sheets may not take the form of a sharp boundary between regions at the melting point, in which sliding is permitted, and regions below that temperature, in which there is no slip. Such a hard switch leads to the paradox of the bed naturally wanting to refreeze as soon as sliding has commenced. A potential alternative structure is a region of subtemperate sliding. Here temperatures are marginally below the melting point and sliding velocities slower than they would if the bed was fully temperate. Rather than being controlled by a standard sliding law, sliding velocities are then constrained by the need to maintain energy balance. This thermal structure arises in temperature-dependent sliding laws in the limit of strong sensitivity to temperature. Here, we analyse the stability of such subtemperate regions, showing that they are subject to a set of instabilities that occur at all length scales between ice thickness and ice sheet length. The fate of these instabilities is to cause the formation of patches of frozen bed, raising the possibility of highly complicated cold-to-temperate transitions with spatial structures at short length scales that cannot be resolved in large-scale ice sheet simulation codes.
For Part I, see [the authors et al., Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 475, No. 2230, Article ID 20190410, 26 p. (2019; Zbl 1472.86038)].Adaptive projection methods for linear fractional programminghttps://zbmath.org/1472.901362021-11-25T18:46:10.358925Z"Bennani, Ahlem"https://zbmath.org/authors/?q=ai:bennani.ahlem"Benterki, Djamel"https://zbmath.org/authors/?q=ai:benterki.djamel"Grar, Hassina"https://zbmath.org/authors/?q=ai:grar.hassinaSummary: In this paper, we are interested in solving a linear fractional program by two different approaches. The first one is based on interior point methods which makes it possible to solve an equivalent linear program to the linear fractional program. The second one allows us to solve a variational inequalities problem equivalent to the linear fractional program by an efficient projection method. Numerical tests were carried out by the two approaches and a comparative study was carried out. The numerical tests show clearly that interior point methods are more efficient than of projection one.Spatially varying multifeedback for robust signalinghttps://zbmath.org/1472.920592021-11-25T18:46:10.358925Z"Wan, Frederic Y. M."https://zbmath.org/authors/?q=ai:wan.frederic-yui-mingSummary: Elaborate regulatory feedback processes are thought to make biological development robust, that is, resistant to changes induced by sustained genetic or environmental perturbations. How this might be accomplished is still not completely understood, especially when models of some known feedback processes have been shown, some analytically and others by numerical simulations, to be ineffective in promoting robust signaling. By working with nonlocal feedback processes, combinations of two of these ineffective feedback processes were found to ensure robustness as measured by a well-defined robustness index for the Dpp-Tkv (decapentaplegic-thickvein) signaling in the wing imaginal disc of \textit{Drosophila} fruit flies. In this paper, we show that there are also spatially varying multifeedback combinations of individually ineffective components that are similarly successful. The analysis of these Hill function type multifeedback models are found to be much more challenging mathematically as the relevant boundary-value problem remains nonlinear even for systems in a steady state of low receptor occupancy.Chemotaxis induced complex dynamics in a novel viral infection modelhttps://zbmath.org/1472.920652021-11-25T18:46:10.358925Z"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.15|wang.wei.3|wang.wei.2|wang.wei.1|wang.wei.12|wang.wei.30|wang.wei.29|wang.wei.9|wang.wei.8|wang.wei.24|wang.wei.16|wang.wei.17|wang.wei.20|wang.wei.26|wang.wei.28|wang.wei.23|wang.wei.13|wang.wei.5|wang.wei.25|wang.wei.18|wang.wei.27|wang.wei.19|wang.wei.21"Zhou, Mengchen"https://zbmath.org/authors/?q=ai:zhou.mengchenSummary: In this letter, we establish a viral infection model with the effect of pyroptosis and chemotaxis. Choosing chemotaxis coefficient as the bifurcation parameter, we derive the necessary condition for the existence of Turing instability. We observe that chemotaxis can induce Turing instability, which may lead to steady state bifurcation or Hopf bifurcation, and spatial patterns.Mathematical modeling of atherogenesis: atheroprotective role of HDLhttps://zbmath.org/1472.920782021-11-25T18:46:10.358925Z"Abi Younes, G."https://zbmath.org/authors/?q=ai:abi-younes.g"El Khatib, N."https://zbmath.org/authors/?q=ai:el-khatib.noaman-a-f|el-khatib.naderSummary: Atherosclerosis is a chronic inflammatory cardiovascular disease in which arteries harden through the build-up of plaques. This work is devoted to the mathematical modeling and analysis of the inflammatory process of atherosclerosis. We propose a mathematical model formed by three coupled partial differential equations of reaction-diffusion type. We take into account three key-role players: the inflammatory immune cells, the inflammatory cytokines and the oxidized low density lipoproteins. A stability analysis of the kinetic system is performed. It leads to the presence of three stable fixed points relevant to appropriate biological states of atherogenesis; no inflammation, stabilized inflammation (stable plaque) and advanced inflammation (vulnerable plaque). The cases that may occur are subject to the variation of the parameters values. A detailed discussion showing how the model fits the biological phenomena is then established. We investigate as well the existence of solutions of traveling waves type along with numerical simulations that show the wave propagation in different cases. This shows that the inflammatory process propagates inside the intima as a traveling wave. Then, we consider the effect of high density lipoprotein (HDL) on the atherosclerotic plaque formation. To do that, we elaborate a map that determines the level of risk of plaque formation with respect to the prevalence of HDL in the blood. These results confirm but also generalize previous results published in the literature. They also give a deeper understanding to the propagation of the inflammation inside the artery in terms of the interplay among the different main players in the whole process.On a subdiffusive tumour growth model with fractional time derivativehttps://zbmath.org/1472.920802021-11-25T18:46:10.358925Z"Fritz, Marvin"https://zbmath.org/authors/?q=ai:fritz.marvin"Kuttler, Christina"https://zbmath.org/authors/?q=ai:kuttler.christina"Rajendran, Mabel L."https://zbmath.org/authors/?q=ai:rajendran.mabel-lizzy"Wohlmuth, Barbara"https://zbmath.org/authors/?q=ai:wohlmuth.barbara-i"Scarabosio, Laura"https://zbmath.org/authors/?q=ai:scarabosio.lauraSummary: In this work, we present and analyse a system of coupled partial differential equations, which models tumour growth under the influence of subdiffusion, mechanical effects, nutrient supply and chemotherapy. The subdiffusion of the system is modelled by a time fractional derivative in the equation governing the volume fraction of the tumour cells. The mass densities of the nutrients and the chemotherapeutic agents are modelled by reaction diffusion equations. We prove the existence and uniqueness of a weak solution to the model via the Faedo-Galerkin method and the application of appropriate compactness theorems. Lastly, we propose a fully discretized system and illustrate the effects of the fractional derivative and the influence of the fractional parameter in numerical examples.On a non-isothermal Cahn-Hilliard model for tumor growthhttps://zbmath.org/1472.920822021-11-25T18:46:10.358925Z"Ipocoana, Erica"https://zbmath.org/authors/?q=ai:ipocoana.ericaSummary: We introduce here a new diffuse interface thermodynamically consistent non-isothermal model for tumor growth in presence of a nutrient in a domain \(\Omega \subset \mathbb{R}^3\). In particular our system describes the growth of a tumor surrounded by healthy tissues, taking into account changes of temperature, proliferation of cells, nutrient consumption and apoptosis. Our aim consists in proving an existence result for our problem associated to the entropy formulation.The existence and numerical method for a free boundary problem modeling the ductal carcinoma in situhttps://zbmath.org/1472.920862021-11-25T18:46:10.358925Z"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.dan"Liu, Keji"https://zbmath.org/authors/?q=ai:liu.keji"Xu, Dinghua"https://zbmath.org/authors/?q=ai:xu.dinghuaSummary: We consider a nonlinear free boundary system for the nutrient concentration, which comes from the mathematical model of ductal carcinoma in situ (DCIS). The motion of the free boundary is given by an integro-differential equation. The existence and uniqueness of the incomplete radial solution for a finite time interval is established, and a novel effective numerical method for simulating the growth of tumor is developed. Results are supported by numerical experiments.Stability of stationary solutions for the glioma growth equations with radial or axial symmetrieshttps://zbmath.org/1472.920872021-11-25T18:46:10.358925Z"Polovinkina, Marina V."https://zbmath.org/authors/?q=ai:polovinkina.marina-v"Debbouche, Amar"https://zbmath.org/authors/?q=ai:debbouche.amar"Polovinkin, Igor P."https://zbmath.org/authors/?q=ai:polovinkin.igor-p"David, Sergio A."https://zbmath.org/authors/?q=ai:david.sergio-adrianiSummary: We investigate a class of nonlinear time-partial differential equations describing the growth of glioma cells. The main results show sufficient conditions for the stability of stationary solutions for these kind of equations. More precisely, we study different spatial variables involving radial or axial symmetries. In addition, we also numerically simulate the system based on three distinct scenarios by considering symmetry across all spatial variables. The numerical results confirm the presence of possible stable states.Well-posedness of a mathematical model of diabetic atherosclerosishttps://zbmath.org/1472.920892021-11-25T18:46:10.358925Z"Xie, Xuming"https://zbmath.org/authors/?q=ai:xie.xumingSummary: Atherosclerosis is a leading cause of death in the United States and worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 21 million Americans have diabetes, a disease where patients' cells cannot efficiently take in dietary sugar, causing it to build up in the blood. In part because diabetes increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. Past work has shown that hyperglycemia and insulin resistance alter function of multiple cell types, including endothelium, smooth muscle cells and platelets, indicating the extent of vascular disarray in this disease. Although the pathophysiology of diabetic vascular disease is generally understood, there is no mathematical model to date that includes the effec