Recent zbMATH articles in MSC 35Bhttps://www.zbmath.org/atom/cc/35B2021-04-16T16:22:00+00:00WerkzeugCorrector equations in fluid mechanics: effective viscosity of colloidal suspensions.https://www.zbmath.org/1456.761342021-04-16T16:22:00+00:00"Duerinckx, Mitia"https://www.zbmath.org/authors/?q=ai:duerinckx.mitia"Gloria, Antoine"https://www.zbmath.org/authors/?q=ai:gloria.antoineSummary: Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result for when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations.Curved fronts in a shear flow: case of combustion nonlinearities.https://www.zbmath.org/1456.351162021-04-16T16:22:00+00:00"El Smaily, Mohammad"https://www.zbmath.org/authors/?q=ai:smaily.mohammad-el|el-smaily.mohammad-ibrahimGradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite-dimensional convex domains.https://www.zbmath.org/1456.280082021-04-16T16:22:00+00:00"Angiuli, L."https://www.zbmath.org/authors/?q=ai:angiuli.luciana"Ferrari, S."https://www.zbmath.org/authors/?q=ai:ferrari.stefania|ferrari.simone|ferrari.stefano|ferrari.sara|ferrari.silvia-l-de-paula|ferrari.silvia"Pallara, D."https://www.zbmath.org/authors/?q=ai:pallara.diegoSummary: Let \(X\) be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure \(\gamma\), and let \(\lambda _1\) be the maximum eigenvalue of the covariance operator associated with \(\gamma\). The associated Cameron-Martin space is denoted by \(H\). For a sufficiently regular convex function \(U:X\rightarrow{{\mathbb{R}}}\) and a convex set \(\Omega \subseteq X\), we set \(\nu :=\mathrm{e}^{-U}\gamma\) and we consider the semigroup \((T_\Omega (t))_{t\ge 0}\) generated by the self-adjoint operator defined via the quadratic form \[(\varphi ,\psi )\mapsto \int _\Omega{\left\langle D_H\varphi ,D_H\psi \right\rangle }_H \mathrm{d}\nu , \] where \(\varphi ,\psi\) belong to \(D^{1,2}(\Omega ,\nu )\), the Sobolev space defined as the domain of the closure in \(L^2(\Omega ,\nu )\) of \(D_H\), the gradient operator along the directions of \(H\). A suitable approximation procedure allows us to prove some pointwise gradient estimates for \((T_{\Omega }(t))_{t\ge 0}\). In particular, we show that \[ |D_H T_{\Omega }(t)f|_H^p\le \mathrm{e}^{- p \lambda _1^{-1} t}(T_{\Omega }(t)|D_H f|^p_H), \quad \, t>0,\ \nu \text{-a.e. in }{\Omega }, \] for any \(p\in [1,+\infty )\) and \(f\in D^{1,p}({\Omega },\nu )\). We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in \({\Omega }\) for the measure \(\nu\) and some improving summability properties for \((T_\Omega (t))_{t\ge 0}\). In addition, we prove that if \(f\) belongs to \(L^p(\Omega ,\nu )\) for some \(p\in (1,\infty )\), then \[ |D_H T_\Omega (t)f|^p_H \le K_p t^{-\frac{p}{2}} T_\Omega (t)|f|^p,\quad \, t>0,\ \nu \text{-a.e. in }\Omega , \] where \(K_p\) is a positive constant depending only on \(p\). Finally, we investigate on the asymptotic behaviour of the semigroup \((T_{\Omega }(t))_{t\ge 0}\) as \(t\) goes to infinity.Asymptotically stable stationary solutions of the reaction-diffusion-advection equation with discontinuous reaction and advection terms.https://www.zbmath.org/1456.350142021-04-16T16:22:00+00:00"Levashova, N. T."https://www.zbmath.org/authors/?q=ai:levashova.natalia-t"Nefedov, N. N."https://www.zbmath.org/authors/?q=ai:nefedov.nikolai-nikolaevich"Nikolaeva, O. A."https://www.zbmath.org/authors/?q=ai:nikolaeva.olga-aThis paper deals with the stationary boundary problem:
\[
\epsilon\frac{d^2u}{dx^2}=A(u,x)\frac{du}{dx}+f(u,x),\ x\in (-1,1),\ u(-1)=u^-,\ u(1)=u^+,\tag{1}
\]
when the advection and reaction terms have a discontinuity of the first type at an interior point \(x_o\in ]-1,1[\).
Several strong assumptions are proposed. Then, the existence of a solution \(u_{\epsilon}\) to problem (1) is proved as \(\epsilon\) is small enough, and an uniform asymptotic approximation of \(u_{\epsilon}\), with an accuracy of order \(O(\epsilon^3)\), is constructed.
Then, it is proved that, given \(\epsilon\) small enough, \(u_{\epsilon}\) is locally unique, and is asymptotically Lyapunov stable, with a suitable stability domain.
Reviewer: Denise Huet (Nancy)Blow-up of solutions for a parabolic Kirchhoff type equation with logarithmic nonlinearity.https://www.zbmath.org/1456.350502021-04-16T16:22:00+00:00"Pişkin, Erhan"https://www.zbmath.org/authors/?q=ai:piskin.erhan"Cömert, Tuğrul"https://www.zbmath.org/authors/?q=ai:comert.tugrulSummary: This study deals with the parabolic type Kirchhoff equation with
logarithmic nonlinearity in a bounded domain. We obtain the finite time blow-up of solutions. This improves and extends some previous studies.Global existence of Landau-Lifshitz-Gilbert equation and self-similar blowup of harmonic map heat flow on \(\mathbb{S}^2\).https://www.zbmath.org/1456.350672021-04-16T16:22:00+00:00"Zhong, Penghong"https://www.zbmath.org/authors/?q=ai:zhong.penghong"Yang, Ganshan"https://www.zbmath.org/authors/?q=ai:yang.ganshan"Ma, Xuan"https://www.zbmath.org/authors/?q=ai:ma.xuanSummary: Under the plane wave setting, the existence of small Cauchy data global solution (or local solution) of Landau-Lifshitz-Gilbert equation is proved. Some variable separation type solutions (include some small data global solution) and self-similar type solutions are constructed for the Harmonic map heat flow on \(\mathbb{S}^2\). As far as we know, there is not any literature that presents the exact blowup solution of this equation. Some explicit solutions which include some finite time gradient-blowup solutions are provided. These blowup examples indicate a finite time blowup will happen in any spacial dimension.Global boundedness and Hölder regularity of solutions to general \(p(x,t)\)-Laplace parabolic equations.https://www.zbmath.org/1456.351192021-04-16T16:22:00+00:00"Ding, Mengyao"https://www.zbmath.org/authors/?q=ai:ding.mengyao"Zhang, Chao"https://www.zbmath.org/authors/?q=ai:zhang.chao.1"Zhou, Shulin"https://www.zbmath.org/authors/?q=ai:zhou.shulinThis paper deals with a class of nonlinear parabolic problems driven by a differential operator with variable exponent. The main results of this paper establish the global boundedness of solutions and Hölder regularity properties of solutions.
Reviewer: Vicenţiu D. Rădulescu (Craiova)Global solutions to a nonlocal Fisher-KPP type problem.https://www.zbmath.org/1456.351082021-04-16T16:22:00+00:00"Bian, Shen"https://www.zbmath.org/authors/?q=ai:bian.shenSummary: We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term \(u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) \), where \(\varOmega\) is a bounded domain in \(\mathbb{R}^{n}(n \geq1)\). The energy method is applied to prove the global existence of the solutions and the results show that the long time behavior of solutions heavily depends on the choice of \(\alpha\), \(\beta\). More precisely, for \(1 \leq\alpha <1+ ( 1-2/p ) \beta\), where \(p\) is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of \(n \geq 3\) and \(\beta=1\), \(\alpha<1+2/n\) is the known Fujita exponent [\textit{H. Fujita}, J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109--124 (1966; Zbl 0163.34002)]. Comparing to Fujita equation [loc. cit.], this paper will give an opposite result to our nonlocal problem.Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints.https://www.zbmath.org/1456.352362021-04-16T16:22:00+00:00"Safdari, Mohammad"https://www.zbmath.org/authors/?q=ai:safdari.mohammadSummary: We prove the optimal \(W^{2, \infty}\) regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be \(C^1\) or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal \(W^{2, \infty}\) regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.Finite-time Euler singularities: a Lagrangian perspective.https://www.zbmath.org/1456.760152021-04-16T16:22:00+00:00"Grafke, Tobias"https://www.zbmath.org/authors/?q=ai:grafke.tobias"Grauer, Rainer"https://www.zbmath.org/authors/?q=ai:grauer.rainerSummary: We address the question of whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows as promising candidates for finite-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, we present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high-resolution adaptively refined numerical simulations and inject Lagrangian tracer particles to monitor geometric properties of vortex line segments. We then verify the assumptions made in the analytical non-blowup criteria introduced by \textit{J. Deng} et al. [Commun. Partial Differ. Equations 31, No. 1--3, 293--306 (2006; Zbl 1158.76305)] connecting vortex line geometry (curvature, spreading) to velocity increase, to rule out singular behavior.Regularity of area minimizing currents mod \(p\).https://www.zbmath.org/1456.490322021-04-16T16:22:00+00:00"De Lellis, Camillo"https://www.zbmath.org/authors/?q=ai:de-lellis.camillo"Hirsch, Jonas"https://www.zbmath.org/authors/?q=ai:hirsch.jonas"Marchese, Andrea"https://www.zbmath.org/authors/?q=ai:marchese.andrea"Stuvard, Salvatore"https://www.zbmath.org/authors/?q=ai:stuvard.salvatoreRegularity of area minimizing currents mod(p) have had an important role in several classical problems of geometric measure theory and mathematical physics. Several authors studied the regularity of area [\textit{C. De Lellis} and \textit{E.N. Spadaro}, Ann. of Math. (2), 183, No. 2, 499--575 (2016; Zbl 1345.49052); \textit{C. De Lellis} and \textit{E.N. Spadaro}, Geom. Funct. Anal., 24, No. 6, 1831--1884 (2014; Zbl 1307.49043); \textit{C. De Lellis} and \textit{E. N. Spadaro}, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14, No. 4, 1239--1269 (2015; Zbl 1343.49073); \textit{T. De Pauw} and \textit{R. Hardt}, Am. J. Math., 134, No. 1, 1--69 (2012; Zbl 1252.49070); \textit{T. De Pauw} and \textit{R. Hardt}, J. Math. Anal. Appl., 418, No. 2, 1047--1061 (2014; Zbl 1347.49073); \textit{L. Simon}, J. Diff. Geom. 38, No. 3, 585--652 (1993; Zbl 0819.53029); \textit{L. Spolaor}, Adv. Math., 350, 747--815 (2019; Zbl 1440.49048)].
The principal objective in this paper is to establish a first general partial regularity theorem for area minimizing currents mod(p), for every p, in any dimension and codimension. More precisely, the authors prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current mod(p) cannot be larger than \(m-1\).
Reviewer: Lakehal Belarbi (Mostaganem)On initial and terminal value problems for fractional nonclassical diffusion equations.https://www.zbmath.org/1456.352222021-04-16T16:22:00+00:00"Tuan, Nguyen Huy"https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan."Caraballo, Tomás"https://www.zbmath.org/authors/?q=ai:caraballo.tomasSummary: In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of fractional pseudo-parabolic equations. For the terminal value problem, we show the well-posedness of our problem in the case \(0<\alpha\leq 1\) and show the ill-posedness in the sense of Hadamard in the case \(\alpha>1\). Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic-type in \(L^q\) norm is first established.Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces.https://www.zbmath.org/1456.352122021-04-16T16:22:00+00:00"Ervin, V. J."https://www.zbmath.org/authors/?q=ai:ervin.vincent-jSummary: In this article we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in \(\mathbb{R}^1\). The analysis is performed in the weighted Sobolev spaces, \( H_{(a, b)}^s(\text{I})\). Three different characterizations of \(H_{(a, b)}^s (\text{I})\) are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters \(\alpha\) and \(r\) defining the fractional diffusion operator. Additionally, the analysis shows that for a sufficiently smooth right hand side function, the regularity of the solution to fractional diffusion reaction equation is lower than that of the fractional diffusion equation. Also, the regularity of the solution to fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation.Nonexistence of positive supersolutions of nonlinear biharmonic equations without the maximum principle.https://www.zbmath.org/1456.352392021-04-16T16:22:00+00:00"Ghergu, Marius"https://www.zbmath.org/authors/?q=ai:ghergu.marius"Taliaferro, Steven D."https://www.zbmath.org/authors/?q=ai:taliaferro.steven-dSummary: We study classical positive solutions of the biharmonic inequality
\[
-\Delta^2 r\geq f(v)
\]
in exterior domains in \(\mathbb{R}^n\) where \(f: (0,\infty) \to (0,\infty)\) is continuous function. We give lower bounds on the growth of \(f(s)\) at \(s=0\) and/or \(s=\infty\) such that inequality \((0.1)\) has no \(C^{4}\) positive solution in any exterior domain of \(\mathbb{R}^n\). Similar results were obtained by Armstrong and Sirakov for \(-\Delta v \geq f(v)\) using a method which depends only on properties related to the maximum principle. Since the maximum principle does not hold for the biharmonic operator, we adopt a different approach which relies on a new representation formula and an a priori pointwise bound for nonnegative solutions of \(-\Delta^{2}u \geq 0\) in a punctured neighborhood of the origin in \(\mathbb{R}^n\).Reaction-diffusion problem in a thin domain with oscillating boundary and varying order of thickness.https://www.zbmath.org/1456.350222021-04-16T16:22:00+00:00"Nakasato, Jean Carlos"https://www.zbmath.org/authors/?q=ai:nakasato.jean-carlos"Pažanin, Igor"https://www.zbmath.org/authors/?q=ai:pazanin.igor"Pereira, Marcone C."https://www.zbmath.org/authors/?q=ai:pereira.marcone-correaSummary: In this paper, we study a reaction-diffusion problem in a thin domain with varying order of thickness. Motivated by the applications, we assume the oscillating behavior of the boundary and prescribe the Robin-type boundary condition simulating the reaction catalyzed by the upper wall. Using the appropriate functional setting and the unfolding operator method, we rigorously derive lower-dimensional approximation of the governing problem. Five different limit problems have been obtained by comparing the magnitude of the reaction mechanism with the variation in domain's thickness.Existence of solution for a class of heat equation involving the \(p(x)\) Laplacian with triple regime.https://www.zbmath.org/1456.351172021-04-16T16:22:00+00:00"Alves, Claudianor O."https://www.zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Boudjeriou, Tahir"https://www.zbmath.org/authors/?q=ai:boudjeriou.tahirSummary: In this paper, we study the local and global existence of solution and the blow-up phenomena for a class of heat equation involving the \(p(x)\)-Laplacian with triple regime.Positive solutions of singular semilinear elliptic equation in bounded NTA-domains.https://www.zbmath.org/1456.350972021-04-16T16:22:00+00:00"Ben Boubaker, Mohamed Amine"https://www.zbmath.org/authors/?q=ai:ben-boubaker.mohamed-amineSummary: We study the existence, the uniqueness and the sharp estimate of a positive solution of the nonlinear equation
\[\Delta v+\psi(\cdot,v)=0,\]
in a bounded NTA-domain \(\Omega\) in \(\mathbb{R}^n \) (\(n\geq 2\)), when a measurable function \(\psi(\cdot,\cdot)\) is continuous and non-increasing with respect to the second variable.Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms.https://www.zbmath.org/1456.350532021-04-16T16:22:00+00:00"Li, Fengbai"https://www.zbmath.org/authors/?q=ai:li.fengbai"Wang, Weike"https://www.zbmath.org/authors/?q=ai:wang.weike"Wang, Yutong"https://www.zbmath.org/authors/?q=ai:wang.yutongSummary: This article concerns the Cauchy problem of conservation laws with nonlocal dissipation-type terms in \(\mathbb{R}^3\). By using Green's function and the time-frequency decomposition method, we study global classical solutions and their long time behavior including pointwise estimates for large initial data, for solutions near the nontrivial equilibrium state.Ingham type approach for uniform observability inequality of the semi-discrete coupled wave equations.https://www.zbmath.org/1456.351312021-04-16T16:22:00+00:00"da Silva Almeida Juniór, Dilberto"https://www.zbmath.org/authors/?q=ai:da-silva-almeida.dilberto-jun"de Jesus Araujo Ramos, Anderson"https://www.zbmath.org/authors/?q=ai:de-jesus-araujo-ramos.anderson"Pantoja Fortes, Joao Carlos"https://www.zbmath.org/authors/?q=ai:pantoja-fortes.joao-carlos"de Lima Santos, Mauro"https://www.zbmath.org/authors/?q=ai:de-lima-santos.mauroSummary: This article concerns an observability inequality for a system of coupled wave equations for the continuous models as well as for the space semi-discrete finite difference approximations. For finite difference and standard finite elements methods on uniform numerical meshes it is known that a numerical pathology produces a blow-up of the constant on the observability inequality as the mesh-size \(h\) tends to zero. We identify this numerical anomaly for coupled wave equations and we prove that there exists a uniform observability inequality in a subspace of solutions generated by low frequencies. We use the Ingham type approach for getting a uniform boundary observability.Maximal regularity for non-autonomous Cauchy problems in weighted spaces.https://www.zbmath.org/1456.350612021-04-16T16:22:00+00:00"Mahdi, Achache"https://www.zbmath.org/authors/?q=ai:mahdi.achache"Hossni, Tebbani"https://www.zbmath.org/authors/?q=ai:hossni.tebbaniSummary: We consider the regularity for the non-autonomous Cauchy problem \[u'(t) + A(t) u(t) = f(t)\quad (t \in [0, \tau]), \quad u(0) = u_0. \] The time dependent operator \(A(t)\) is associated with (time dependent) sesquilinear forms on a Hilbert space \(\mathcal{H}\). We prove the maximal regularity result in temporally weighted \(L^2\)-spaces and other regularity properties for the solution of the problem under minimal regularity assumptions on the forms and the initial value \(u_0\). Our results are motivated by boundary value problems.Symmetric vortices for two-component \(p\)-Ginzburg-Landau systems.https://www.zbmath.org/1456.351902021-04-16T16:22:00+00:00"Duan, Lipeng"https://www.zbmath.org/authors/?q=ai:duan.lipeng"Yang, Jun"https://www.zbmath.org/authors/?q=ai:yang.jun.1|yang.jun.2|yang.jun|yang.jun.3Summary: Given \(p > 2\) for the following coupled \(p\)-Ginzburg-Landau model in \(\mathbb{R}^2\)
\[
\begin{aligned}
-\Delta_p u^+ + \left[A_+ (|u^+|^2 - t^{+^2}) + A_0(|u^-|^2 - t^{-^2})\right] u^+ = 0, \\
-\Delta_p u^- + \left[A_- (|u^-|^2 - t^{-^2}) + A_0(|u^+|^2 - t^{+^2})\right] u^- = 0,
\end{aligned}
\]
with the constraints
\[
A_+, A_- > 0, A_0^2 < A_+ A_- \quad \text{and} \quad t^+, t^- > 0,
\]
we consider the existence of symmetric vortex solutions \(u(x) = (U_p^+(r) e^{in^+ \theta}, U_p^-(r) e^{in^- \theta})\) with given degree \((n^+, n^-) \in \mathbb{Z}^2\), and then prove the uniqueness and regularity results for the vortex profile \((U_p^+, U_p^-)\) under more constraint of the parameters. Moreover, we also establish the stability result for second variation of the energy around this vortex profile when we consider the perturbations in a space of radial functions.Effects of quenched disorder on critical transitions in pattern-forming systems.https://www.zbmath.org/1456.350302021-04-16T16:22:00+00:00"Yizhaq, Hezi"https://www.zbmath.org/authors/?q=ai:yizhaq.hezi"Bel, Golan"https://www.zbmath.org/authors/?q=ai:bel.golanGlobal existence and asymptotic behavior of periodic solutions to a fractional chemotaxis system on the weakly competitive case.https://www.zbmath.org/1456.352172021-04-16T16:22:00+00:00"Lei, Yuzhu"https://www.zbmath.org/authors/?q=ai:lei.yuzhu"Liu, Zuhan"https://www.zbmath.org/authors/?q=ai:liu.zuhan"Zhou, Ling"https://www.zbmath.org/authors/?q=ai:zhou.lingSummary: In this paper, we consider a two-species parabolic-parabolic-elliptic chemotaxis system with weak competition and a fractional diffusion of order \(s\in(0,2)\). It is proved that for \(s>2p_0\), where \(p_0\) is a nonnegative constant depending on the system's parameters, there admits a global classical solution. Apart from this, under the circumstance of small chemotactic strengths, we arrive at the global asymptotic stability of the coexistence steady state.A Poincare's inequality with non-uniformly degenerating gradient.https://www.zbmath.org/1456.260182021-04-16T16:22:00+00:00"Mamedov, Farman"https://www.zbmath.org/authors/?q=ai:mamedov.farman-imranMain goal of the author is to show a Poincaré type \(p - q\) inequality in a homogeneous space \((\mathbb{R}^N, d,\mu)\) estimating the weighted Lebesgue norm of a Lipschitz continuous function \(f\) on a bounded convex domain \(\Omega \subset \mathbb{R}^N\) via the Lebesgue norm of its non-uniformly degenerating gradient.
Then, the author studies regularity properties of weak solutions of second-order semi-elliptic equations
\(\operatorname{div} (A(x)\nabla u) = 0\)
with matrix \(A(x) \geq 0\).
The general results obtained in the first two theorems allow the author to claim new assertions on the Poincaré-Sobolev inequality and he further studies equations containing a Grushin-type operator.
Reviewer: Maria Alessandra Ragusa (Catania)Viscous conservation laws in 1D with measure initial data.https://www.zbmath.org/1456.351012021-04-16T16:22:00+00:00"Bank, Miriam"https://www.zbmath.org/authors/?q=ai:bank.miriam"Ben-Artzi, Matania"https://www.zbmath.org/authors/?q=ai:ben-artzi.matania"Schonbek, Maria E."https://www.zbmath.org/authors/?q=ai:schonbek.maria-elenaSummary: The one-dimensional viscous conservation law is considered on the whole line
\[
u_t+f(u)_x=\varepsilon u_{xx},\quad (x,t)\in\mathbb{R}\times \overline{\mathbb{R}_+},\quad \varepsilon>0,
\]
subject to positive measure initial data.
The flux \(f\in C^1(\mathbb{R})\) is assumed to satisfy a \(p\)-condition, a weak form of convexity. In particular, any flux of the form \(f(u)=\sum_{i=1}^Ja_iu^{m_i}\) is admissible if \(a_i>0\), \(m_i>1\), \(i=1,2,\dots,J\).
The only case treated hitherto in the literature is \(f(u)=u^m\) [\textit{M. Escobedo} et al., Arch. Ration. Mech. Anal. 124, No. 1, 43--65 (1993; Zbl 0807.35059)] and the initial data is a ``single source'', namely, a multiple of the delta function. The corresponding solutions have been labeled as ``source-type'' and the treatment made substantial use of the special form of both the flux and the initial data. In this paper existence and uniqueness of solutions is established. The method of proof relies on sharp decay estimates for the viscous Hamilton-Jacobi equation. Some estimates are independent of the viscosity coefficient, thus leading to new estimates for the (inviscid) hyperbolic conservation law.Large solutions to elliptic equations involving fractional Laplacian.https://www.zbmath.org/1456.352112021-04-16T16:22:00+00:00"Chen, Huyuan"https://www.zbmath.org/authors/?q=ai:chen.huyuan"Felmer, Patricio"https://www.zbmath.org/authors/?q=ai:felmer.patricio-l"Quaas, Alexander"https://www.zbmath.org/authors/?q=ai:quaas.alexanderSummary: The purpose of this paper is to study boundary blow up solutions for semi-linear fractional elliptic equations of the form\[(0.1)\begin{cases} (- \operatorname{\Delta})^\alpha u(x) + | u |^{p - 1} u(x) = f(x), \quad & x \in \Omega, \\ u(x) = 0, & x \in \overline{\Omega}^c, \\ \lim\limits_{x \in \Omega, x \to \partial \Omega} u(x) = + \infty, \end{cases}\] where \(p > 1\), \(\Omega\) is an open bounded \(C^2\) domain of \(\mathbb{R}^N\), \(N \geq 2\), the operator \((- \operatorname{\Delta})^\alpha\) with \(\alpha \in(0, 1)\) is the fractional Laplacian and \(f : \Omega \to \mathbb{R}\) is a continuous function which satisfies some appropriate conditions. We obtain that problem \((0.1)\) admits a solution with boundary behavior like \(d(x)^{- \frac{2 \alpha}{p - 1}}\), when \(1 + 2 \alpha < p < 1 - \frac{2 \alpha}{\tau_0(\alpha)}\), for some \(\tau_0(\alpha) \in(- 1, 0)\), and has infinitely many solutions with boundary behavior like \(d(x)^{\tau_0(\alpha)}\), when \(\max \{1 - \frac{2 \alpha}{\tau_0} + \frac{\tau_0(\alpha) + 1}{\tau_0}, 1 \} < p < 1 - \frac{2 \alpha}{\tau_0}\). Moreover, we also obtained some uniqueness and non-existence results for problem \((0.1)\).Solutions to viscous Burgers equations with time dependent source term.https://www.zbmath.org/1456.350772021-04-16T16:22:00+00:00"Engu, Satyanarayana"https://www.zbmath.org/authors/?q=ai:engu.satyanarayana"Sahoo, Manas R."https://www.zbmath.org/authors/?q=ai:sahoo.manas-ranjan"Berke, Venkatramana P."https://www.zbmath.org/authors/?q=ai:berke.venkatramana-pSummary: We study the existence and uniqueness of weak solutions for a Cauchy problem of a viscous Burgers equation with a time dependent reaction term involving Dirac measure. After applying a Hopf like transformation, we investigate the associated two initial boundary value problems by assuming a common boundary. The existence of the boundary data is shown with the help of Abel's integral equation. We then derive explicit representation of the boundary function. Also, we prove that the solutions of associated initial boundary value problems converge uniformly to a nonzero constant on compact sets as \(t\) approaches \(\infty\).Remarks on Liouville type theorems for the 3D stationary MHD equations.https://www.zbmath.org/1456.351642021-04-16T16:22:00+00:00"Li, Zhouyu"https://www.zbmath.org/authors/?q=ai:li.zhouyu"Liu, Pan"https://www.zbmath.org/authors/?q=ai:liu.pan"Niu, Pengcheng"https://www.zbmath.org/authors/?q=ai:niu.pengchengSummary: The aim of this paper is to establish Liouville type results for the stationary MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero. Moreover, we also obtain Liouville type theorem for the case of axially symmetric MHD equations. Our results generalize previous works by Schulz [14] and \textit{G. Seregin} [Nonlinearity 29, No. 8, 2191--2195 (2016; Zbl 1350.35047)] and \textit{W. Wang} [St. Petersbg. Math. J. 31, No. 2, 387--393 (2020; Zbl 1434.35063)].Adaptive regularisation for ensemble Kalman inversion.https://www.zbmath.org/1456.650952021-04-16T16:22:00+00:00"Iglesias, Marco"https://www.zbmath.org/authors/?q=ai:iglesias.marco-a"Yang, Yuchen"https://www.zbmath.org/authors/?q=ai:yang.yuchenMeromorphic solutions of generalized inviscid Burgers' equations and related PDEs.https://www.zbmath.org/1456.350092021-04-16T16:22:00+00:00"Lü, Feng"https://www.zbmath.org/authors/?q=ai:lu.fengSummary: The purposes of this paper are twofold. The first one is to describe entire solutions of certain type of PDEs in \(\mathbb{C}^n\) with the modified KdV-Burgers equation and modified Zakharov-Kuznetsov equation as the prototypes. The second one is to characterize entire and meromorphic solutions of generalized inviscid Burgers' equations in \(\mathbb{C}^2\).Determination of the reaction coefficient in a time dependent nonlocal diffusion process.https://www.zbmath.org/1456.650922021-04-16T16:22:00+00:00"Ding, Ming-Hui"https://www.zbmath.org/authors/?q=ai:ding.minghui"Zheng, Guang-Hui"https://www.zbmath.org/authors/?q=ai:zheng.guanghuiWeak well-posedness of multidimensional stable driven SDEs in the critical case.https://www.zbmath.org/1456.601432021-04-16T16:22:00+00:00"Chaudru de Raynal, Paul-Éric"https://www.zbmath.org/authors/?q=ai:de-raynal.paul-eric-chaudru"Menozzi, Stéphane"https://www.zbmath.org/authors/?q=ai:menozzi.stephane"Priola, Enrico"https://www.zbmath.org/authors/?q=ai:priola.enricoLie symmetry analysis and similarity solutions for the Camassa-Choi equations.https://www.zbmath.org/1456.351662021-04-16T16:22:00+00:00"Paliathanasis, Andronikos"https://www.zbmath.org/authors/?q=ai:paliathanasis.andronikosSummary: The method of Lie symmetry analysis of differential equations is applied to determine exact solutions for the Camassa-Choi equation and its generalization. We prove that the Camassa-Choi equation is invariant under an infinity-dimensional Lie algebra, with an essential five-dimensional Lie algebra. The application of the Lie point symmetries leads to the construction of exact similarity solutions.A nonlocal Dirichlet problem with impulsive action: estimates of the growth for the solutions.https://www.zbmath.org/1456.350322021-04-16T16:22:00+00:00"Ferreira, Jaqueline da Costa"https://www.zbmath.org/authors/?q=ai:da-costa-ferreira.jaqueline"Pereira, Marcone Corrêa"https://www.zbmath.org/authors/?q=ai:pereira.marcone-correaSummary: Through this paper we deal with the asymptotic behaviour as \(t\rightarrow +\infty\) of the solutions for a nonlocal diffusion problem with impulsive actions and Dirichlet condition. We establish a decay rate for the solutions assuming appropriate hypotheses on the impulsive functions and the nonlinear reaction.Minimizers of convex functionals with small degeneracy set.https://www.zbmath.org/1456.490292021-04-16T16:22:00+00:00"Mooney, Connor"https://www.zbmath.org/authors/?q=ai:mooney.connorLet \(F:\mathbb R^n\to\mathbb R\) be convex and of class \(C^1\).
The paper studies the regularity of Lipschitz minimizers of
\[
E(u)=\int_{B_1}F(\nabla u)\,dx
\]
in \(\mathbb R^n\), i.e. functions \(u\in W^{1, \infty}(B_1)\) satisfying \(E(u+\varphi)\ge E(u)\) for all \(\varphi\in C^1_0(B_1)\).
In the extreme case that the graph of \(F\) contains a line segment, minimizers are no better than Lipschitz by simple examples. In the other extreme that \(F\) is smooth and uniformly convex, De Giorgi and Nash proved that Lipschitz minimizers are smooth and solve the Euler-Lagrange equation \(F_{ij}(\nabla u )u_{ij}=0\) classically [\textit{E. De Giorgi}, Mem. Accad. Sci. Torino, P. I., III. Ser. 3, 25--43 (1957; Zbl 0084.31901); \textit{J. F. Nash}, Am. J. Math. 80, 931--954 (1958; Zbl 0096.06902)]. The paper examines the intermediate case where \(F\) is strictly convex, but the eigenvalues of \(D^2F\) go to 0 or \(\infty\) on some set \(D_F\). Such functionals arise naturally in the study of anisotropic surface tensions [\textit{M. G. Delgadino} et al., Arch. Ration. Mech. Anal. 230, No. 3, 1131--1177 (2018; Zbl 1421.35076)], traffic flow [\textit{M. Colombo} and \textit{A. Figalli}, J. Math. Pures Appl. (9) 101, No. 1, 94--117 (2014; Zbl 1282.35175)], and statistical mechanics [\textit{H. Cohn} et al., J. Am. Math. Soc. 14, No. 2, 297--346 (2001; Zbl 1037.82016); \textit{R. Kenyon} et al., Ann. Math. (2) 163, No. 3, 1019--1056 (2006; Zbl 1154.82007)].
More precisely, the author assumes that there is a compact subset \(D_F\) of \( \mathbb R^n\) such that
\[
F\in C^2(\mathbb R^n\setminus D_F),\quad D_F=\mathbb R^n\setminus\left(\cup_k\{k^{-1}I<D^2F<kI\}\right).
\]
Theorem 2.1 states that if \(D_F\) is finite and is contained in a two-dimensional affine subspace of \(\mathbb R^n\), then \(u\in C^1(B_1)\). Theorem 2.3 shows that in general a Lipschitz minimizer of \(E\) may not be of class \(C^1\), by constructing a singular Lipschitz minimizer in \(\mathbb R^4\).
The above leave opens the possibility that Lipschitz minimizers are \(C^1\) in dimension \(n\ge 3\) in the case where \(D_F\) consists of finitely many points: the problem is connected to a result of Alexandrov in the classical differential geometry of convex surfaces and a related counterexample is conjectured (Conjecture 2.4).
Reviewer: Carlo Mariconda (Padova)Regularity for degenerate two-phase free boundary problems.https://www.zbmath.org/1456.352352021-04-16T16:22:00+00:00"Leitão, Raimundo"https://www.zbmath.org/authors/?q=ai:leitao.raimundo-a"de Queiroz, Olivaine S."https://www.zbmath.org/authors/?q=ai:de-queiroz.olivaine-santana"Teixeira, Eduardo V."https://www.zbmath.org/authors/?q=ai:teixeira.eduardo-v-oSummary: We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase free boundary problems, \(\mathcal{J}_\gamma \to \min\), ruled by nonlinear, \(p\)-degenerate elliptic operators. Included in such family are heterogeneous cavitation problems of Prandtl-Batchelor type, singular degenerate elliptic equations; and obstacle type systems. The Euler-Lagrange equation associated to \(\mathcal{J}_\gamma\) becomes singular along the free interface \(\{u = 0 \}\). The degree of singularity is, in turn, dimmed by the parameter \(\gamma \in [0, 1]\). For \(0 < \gamma < 1\) we show that local minima are locally of class \(C^{1, \alpha}\) for a sharp \(\alpha\) that depends on dimension, \(p\) and \(\gamma\). For \(\gamma = 0\) we obtain a quantitative, asymptotically optimal result, which assures that local minima are Log-Lipschitz continuous. The results proven in this article are new even in the classical context of linear, nondegenerate equations.Structural stability of supersonic solutions to the Euler-Poisson system.https://www.zbmath.org/1456.351362021-04-16T16:22:00+00:00"Bae, Myoungjean"https://www.zbmath.org/authors/?q=ai:bae.myoungjean"Duan, Ben"https://www.zbmath.org/authors/?q=ai:duan.ben"Xiao, Jingjing"https://www.zbmath.org/authors/?q=ai:xiao.jingjing"Xie, Chunjing"https://www.zbmath.org/authors/?q=ai:xie.chunjingThe authors study the steady Euler-Poisson system for a flow of charged media and look for smooth supersonic solutions. The domain is a 2D rectangle, and the boundary data are of Dirichlet type. Well-posedness of the problem for the irrotational flow and small perturbations of initial and boundary data is proved.
Reviewer: Ilya A. Chernov (Petrozavodsk)Finite vs infinite derivative loss for abstract wave equations with singular time-dependent propagation speed.https://www.zbmath.org/1456.351422021-04-16T16:22:00+00:00"Ghisi, Marina"https://www.zbmath.org/authors/?q=ai:ghisi.marina"Gobbino, Massimo"https://www.zbmath.org/authors/?q=ai:gobbino.massimoSummary: We consider an abstract wave equation with a propagation speed that depends only on time. We investigate well-posedness results with finite derivative loss in the case where the propagation speed is smooth for positive times, but potentially singular at the initial time.
We prove that solutions exhibit a finite derivative loss under a family of conditions that involve the blow up rate of the first and second derivative of the propagation speed, in the spirit that the weaker is the requirement on the first derivative, the stronger is the requirement on the second derivative. Our family of conditions interpolates between the two limit cases that were already known in the literature.
We also provide the counterexamples that show that, as soon as our conditions fail, solutions can exhibit an infinite derivative loss. The existence of such pathologies was an open problem even in the two extreme cases.Long-range memory effects in a magnetized Hindmarsh-Rose neural network.https://www.zbmath.org/1456.828032021-04-16T16:22:00+00:00"Etémé, Armand S."https://www.zbmath.org/authors/?q=ai:eteme.armand-sylvin"Tabi, Conrad B."https://www.zbmath.org/authors/?q=ai:tabi.conrad-bertrand"Mohamadou, Alidou"https://www.zbmath.org/authors/?q=ai:mohamadou.alidou"Kofané, Timoléon C."https://www.zbmath.org/authors/?q=ai:kofane.timoleon-crepinThe article refers to the Hindmarsh-Rose model introduced in a paper of \textit{A. S. Etémé} et al. [Commun. Nonlinear Sci. Numer. Simul. 72, 432--440 (2019; Zbl 07264754)]. Since one intends to investigate the partnership between intrinsical memristive and boundary long-range couplings on modulated waves along a chain made of \(N\) identical neurons, long-range (LR) diffusive effects are added. The model under study has the form
$$\dot{x_n}=y_n-ax^3_n +bx^2_n-z_n-{k_1}w(\phi_n)x_n+\sum_{j=1}^{J}{ D_j}(s)(x_{n+j}-2x_n+x_{n-j}),$$
$$\dot{y_n}=c-dx^2_n-ey_n,\dot{z_n}={\Omega^2}_0(x_n-x_e)-rz_n,$$
$$\dot{\phi}_n=x_n-k_2\phi_n+\phi_0.$$
By applying a multi-scaling approach, the system is reduced to a discrete nonlinear Schrödinger equation with the coefficients depending on some control parameters. One investigates analytically the modulation instability (MI) and different features of MI. The instability criterion and the critical amplitude expression are obtained. An extensive part of the article is devoted to numerical experiments on: MI and chaotic behavior of neural network, MI and neural network synchronicity, MI and characterization of chimera patterns. Comments and future trends can be found at the end of article.
Reviewer: Claudia Simionescu-Badea (Wien)Asymptotic analysis of an advection-diffusion equation and application to boundary controllability.https://www.zbmath.org/1456.352042021-04-16T16:22:00+00:00"Amirat, Youcef"https://www.zbmath.org/authors/?q=ai:amirat.youcef-ait"Münch, Arnaud"https://www.zbmath.org/authors/?q=ai:munch.arnaudThe authors consider the advection-diffusion equation
\[
y^{\varepsilon}_t-\varepsilon y^{\varepsilon}_{xx}+My^{\varepsilon}_x=0,\quad (x,t)\in (0,1)\times(0,T),
\]
\[
y^{\varepsilon}(0,t)=v^{\varepsilon}(t),\quad y^{\varepsilon}(1,t)=0,\quad t\in (0,T),\quad (1)
\]
\[
y^{\varepsilon}(x.0)=y^{\varepsilon}_0(x)\quad x\in (0,1),
\]
where \(\varepsilon\) is the diffusion coefficient, \(M\) is the transport coefficient, \(v^{\varepsilon}\)=\(v^{\varepsilon}(t)\) is the control function, \(y^{\varepsilon}_0\) is the initial data, and \(y^{\varepsilon}=y^{\varepsilon}(x,t)\) is the associated state.
The main purpose of the article is to perform the asymptotic analysis of (1) for the case \(M>0\), assuming \(v^{\epsilon}\) fixed and satisfying compatibility conditions at the initial time \(t=0\) with the initial condition \(y^{\varepsilon}_0\) as \(x=0\).
Supposing that the initial condition does not depend on \(\varepsilon\) and that the control function \(v^{\varepsilon}\) is given in the form \(v^{\varepsilon}=\sum\limits_{k=0}^m \varepsilon^k v^k\), the authors construct the accurate asymptotic approximation \(w^{\varepsilon}\) of the solution \(y^{\varepsilon}\) by using the method of matched asymptotic expansions. The inner and outer solutions are determined by using explicit formulae, and \(w^{\varepsilon}_m\) is obtained by combining the inner and outer solutions through a process of matching. It is proved that \(w^{\varepsilon}_m\) is a regular and strong approximation of \(y^{\varepsilon}\), as \(\varepsilon \to 0^{+}\). The error estimate is obtained. The same approach is applied to the solution \(\varphi^{\varepsilon}\) of the adjoint problem
\[
-\varphi^{\varepsilon}_t-\varepsilon \varphi^{\varepsilon}_{xx}-M\varphi^{\varepsilon}_x=0,\quad (x,t)\in (0,1)\times(0,T),
\]
\[
\varphi^{\varepsilon}(0,t)=\varphi^{\varepsilon}(1,t)=0 \quad t\in (0,T)
\]
\[
\varphi^{\varepsilon}(x.T)=\varphi^{\varepsilon}_T(x)\quad x\in (0,1),
\]
where \(\varphi_T^{\varepsilon}\) is a function of the form \(\varphi^{\varepsilon}_T=\sum\limits^{m}_{k=0}\varepsilon^k\varphi^k_T\), the functions \(\varphi^0_T,\varphi^1_T,\ldots,\varphi^m_T\) being given.
The authors show that under some conditions on the functions \(v^k\) and on the initial condition \(y_0\) one can pass to the limit as \(m\to \infty\) and establish a convergence result of the sequence \((w^{\varepsilon}_m)_{(m\ge 0)}\).
Then they use the asymptotic results to discuss the controllability properties of the solution for \(T\ge 1/M\).
Reviewer: Artyom Andronov (Saransk)Local energy decay for the damped Klein-Gordon equation in exterior domain.https://www.zbmath.org/1456.351262021-04-16T16:22:00+00:00"Malloug, Mohamed"https://www.zbmath.org/authors/?q=ai:malloug.mohamedSummary: We prove uniform local energy decay for the solution of the dissipative Klein-Gordon equation on an exterior domain under some geometric condition called ``exterior geometric control''.Green's function for the Schrödinger equation with a generalized point interaction and stability of superoscillations.https://www.zbmath.org/1456.811652021-04-16T16:22:00+00:00"Aharonov, Yakir"https://www.zbmath.org/authors/?q=ai:aharonov.yakir"Behrndt, Jussi"https://www.zbmath.org/authors/?q=ai:behrndt.jussi"Colombo, Fabrizio"https://www.zbmath.org/authors/?q=ai:colombo.fabrizio"Schlosser, Peter"https://www.zbmath.org/authors/?q=ai:schlosser.peterSummary: In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the \(\delta\) and \(\delta^\prime\)-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.Blow-up for Strauss type wave equation with damping and potential.https://www.zbmath.org/1456.350452021-04-16T16:22:00+00:00"Dai, Wei"https://www.zbmath.org/authors/?q=ai:dai.wei.3|dai.wei.5|dai.wei.2|dai.wei.6|dai.wei.4|dai.wei.7|dai.wei"Kubo, Hideo"https://www.zbmath.org/authors/?q=ai:kubo.hideo"Sobajima, Motohiro"https://www.zbmath.org/authors/?q=ai:sobajima.motohiroIn this paper, the authors study the blow-up phenomenon for the Cauchy problem of the wave equation with space-dependent critical damping and potential terms in \(\mathbb{R}^n\) (\(n\ge 1\)), \(\partial_t^2 u -\Delta u + Ar^{-1}\partial_t u +Br^{-2}u=|u|^p\), with
\(B>-(n/2-1)^2\), \(0\leq A < n-1+2\rho\) and \(\rho=\sqrt{(n/2-1)^2+B}-(n/2-1)\). Heuristically, the damping coefficient \(A\) and the potential coefficient \(B\) will lead to certain spatial dimensional shift from \(n\) to \(n+A\) and \(n+\rho\) separately. For small, nontrivial, nonnegative data of size \(\varepsilon\), it is shown that the solutions blow up in finite time for \(1+1 / (n+\rho-1) < p< p_c=\max(p_S(n+A), 1+2/(n+\rho-1))\), if \(p_c>1+1/(n+\rho-1)\). Here \(p_S(k)\) is the positive root of \((k-1)p(p-1)=2(p+1)\) (when \(k>1\), otherwise \(p_S(k)=\infty\)), which is also known as the Strauss exponent. In addition, the expected upper bound of the lifespan is also obtained. In order to prove the blow-up result, the authors employ the test function method. The test functions are based on a family of special solutions, inside the light cone \(|x|< t+\lambda\), to the linear dual problem \(\partial_t^2\Psi - Ar^{-1}\partial_t \Psi -\Delta \Psi +Br^{-2}\Psi=0\), which have the form
\(\Psi_{\beta}(t,x)=r^\rho (t+r+\lambda)^{-\beta}\phi(\frac{2r}{t+r+\lambda})\), (\(\beta\in\mathbb{R}\), \(\lambda\geq 0\)), for hypergeometric functions \(\phi\) depending on \(n\), \(A\), \(B\).
Reviewer: Chengbo Wang (Hangzhou)Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms.https://www.zbmath.org/1456.350492021-04-16T16:22:00+00:00"Palmieri, Alessandro"https://www.zbmath.org/authors/?q=ai:palmieri.alessandro"Takamura, Hiroyuki"https://www.zbmath.org/authors/?q=ai:takamura.hiroyukiSummary: In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type. The proof of the blow-up results is based on an iteration argument. We find as critical curve for the pair of exponents \((p,q)\) in the nonlinear terms the same one found for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.Stability of steady-state for 3-D hydrodynamic model of unipolar semiconductor with ohmic contact boundary in hollow ball.https://www.zbmath.org/1456.829362021-04-16T16:22:00+00:00"Mei, Ming"https://www.zbmath.org/authors/?q=ai:mei.ming"Wu, Xiaochun"https://www.zbmath.org/authors/?q=ai:wu.xiaochun"Zhang, Yongqian"https://www.zbmath.org/authors/?q=ai:zhang.yongqianSummary: The existence of stationary subsonic solutions and their stability for 3-D hydrodynamic model of unipolar semiconductors with the Ohmic contact boundary have been open for long time due to some technical reason, as we know. In this paper, we consider 3-D radial solutions to the system in a hollow ball, and prove that the 3-D radial subsonic stationary solutions uniquely exist and are asymptotically stable, when the initial perturbations around the subsonic steady-state are small enough. Different from the existing studies on the radial solutions for fluid dynamics where the inner boundary of the hollow ball must be far away from the singular origin, here we may allow the chosen inner boundary arbitrarily close to the singular origin and reveal the relationship between the inner boundary and the large time behavior of the radial solution. This partially answers the open question of the stability of stationary waves subjected to the Ohmic contact boundary conditions in the multiple dimensional space. We also prove the existence of non-flat stationary subsonic solution, which essentially improve and develop the previous studies in this subject. The proof is based on the technical energy estimates in certain weighted Sobolev spaces, where the weight functions are artfully selected to be the distance of the targeted spatial location and the singular point.Lipschitz regularity results for a class of obstacle problems with nearly linear growth.https://www.zbmath.org/1456.490282021-04-16T16:22:00+00:00"Bertazzoni, Giacomo"https://www.zbmath.org/authors/?q=ai:bertazzoni.giacomo"Riccò, Samuele"https://www.zbmath.org/authors/?q=ai:ricco.samueleThis paper deals with the regularity of variational obstacle problems with nearly linear growth. The authors prove the Lipschitz continuity of solutions for a large class of problems and then the Lavrentiev phenomenon does not occur.
The main tool used here is a new higher differentiability result which reveals to be crucial because it allows to transform the constrained problem in an unconstrained one, by means a linearization procedure.
It is interesting to note that the same Sobolev regularity both for the gradient of the obstacle and for the coefficients is assumed.
Reviewer: Elvira Mascolo (Firenze)Quasi-periodic solutions for fractional nonlinear Schrödinger equation.https://www.zbmath.org/1456.370802021-04-16T16:22:00+00:00"Xu, Xindong"https://www.zbmath.org/authors/?q=ai:xu.xindongSummary: We establish an infinite dimensional KAM theorem with dense normal frequency. As an application, we use this theorem to study the fractional NLS
\[
iu_t-|\partial_x|^{\frac{1}{2}}u+M_\xi u=\varepsilon f(|u|^2)u,\quad x\in\mathbb{T},t\in\mathbb{R},
\]
where \(f\) is real analytic in a neighborhood of \(0\in \mathbb{C}\) and \(\varepsilon >0\) is small enough. We obtain a family of small-amplitude quasi-periodic solutions with linear stability.Classical solutions to the initial-boundary value problems for nonautonomous fractional diffusion equations.https://www.zbmath.org/1456.352182021-04-16T16:22:00+00:00"Mu, Jia"https://www.zbmath.org/authors/?q=ai:mu.jia"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.6|liu.yang.3|liu.yang.2|liu.yang.17|liu.yang.18|liu.yang.11|liu.yang.8|liu.yang.21|liu.yang.14|liu.yang.22|liu.yang.19|liu.yang.9|liu.yang.4|liu.yang.5|liu.yang.20|liu.yang.12|liu.yang.13|liu.yang|liu.yang.23|liu.yang.10|liu.yang.15|liu.yang.16|liu.yang.1"Zhang, Huanhuan"https://www.zbmath.org/authors/?q=ai:zhang.huanhuanSummary: In this paper, we investigate a class of nonautonomous fractional diffusion equations (NFDEs). Firstly, under the condition of weighted Hölder continuity, the existence and two estimates of classical solutions are obtained by virtue of the properties of the probability density function and the evolution operator family. Secondly, it focuses on the continuity and an estimate of classical solutions in the sense of fractional power norm. The results generalize some existing results on classical solutions and provide theoretical support for the application of NFDE.Nonlinear stability analyses of Turing patterns for a mussel-algae model.https://www.zbmath.org/1456.350282021-04-16T16:22:00+00:00"Cangelosi, Richard A."https://www.zbmath.org/authors/?q=ai:cangelosi.richard-a"Wollkind, David J."https://www.zbmath.org/authors/?q=ai:wollkind.david-j"Kealy-Dichone, Bonni J."https://www.zbmath.org/authors/?q=ai:kealy-dichone.bonni-j"Chaiya, Inthira"https://www.zbmath.org/authors/?q=ai:chaiya.inthiraSummary: A particular interaction-diffusion mussel-algae model system for the development of spontaneous stationary young mussel bed patterning on a homogeneous substrate covered by a quiescent marine layer containing algae as a food source is investigated employing weakly nonlinear diffusive instability analyses. The main results of these analyses can be represented by plots in the ratio of mussel motility to algae lateral diffusion versus the algae reservoir concentration dimensionless parameter space. Regions corresponding to bare sediment and mussel patterns consisting of rhombic or hexagonal arrays and isolated clusters of clumps or gaps, an intermediate labyrinthine state, and homogeneous distributions of low to high density may be identified in this parameter space. Then those Turing diffusive instability predictions are compared with both relevant field and laboratory experimental evidence and existing numerical simulations involving differential flow migrating band instabilities for the associated interaction-dispersion-advection mussel-algae model system as well as placed in the context of the results from some recent nonlinear pattern formation studies.Tasks with fast oscillating data. Two examples of asymptotics construction.https://www.zbmath.org/1456.350192021-04-16T16:22:00+00:00"Ivleva, N. S."https://www.zbmath.org/authors/?q=ai:ivleva.n-sSummary: For two specific problems with rapidly oscillating data in time -- the semilinear parabolic system with two spatial variables and the Navier-Stokes system that simulates the fluid flow in the flat case -- the question of constructing asymptotic expansions of their time-periodic solutions is solved. Both problems are considered in the cylinder, infinite in time, the axis of which is the temporary numerical axis, and the basis is the two-dimensional unit circle. The Dirichlet conditions are taken as boundary conditions. The construction of these asymptotic expansions is based on two algorithms developed, justified and obtained earlier by the author and V. B. Levenstam.Continuous and variable branching asymptotics.https://www.zbmath.org/1456.350602021-04-16T16:22:00+00:00"Hedayat Mahmoudi, M."https://www.zbmath.org/authors/?q=ai:hedayat-mahmoudi.m"Schulze, B.-W."https://www.zbmath.org/authors/?q=ai:schulze.bert-wolfgang"Tepoyan, L."https://www.zbmath.org/authors/?q=ai:tepoyan.liparit|tepoyan.liparit-pSummary: The regularity of solutions to elliptic equations on a manifold with singularities, say, an edge, can be formulated in terms of asymptotics in the distance variable \(r>0\) to the singularity. In simplest form such asymptotics turn to a meromorphic behaviour under applying the Mellin transform on the half-axis. Poles, multiplicity, and Laurent coefficients form a system of asymptotic data which depend on the specific operator. Moreover, these data may depend on the variable \(y\) along the edge. We then have \(y\)-dependent families of meromorphic functions with variable poles, jumping multiplicities and a discontinuous dependence of Laurent coefficients on \(y\). We study here basic phenomena connected with such variable branching asymptotics, formulated in terms of variable continuous asymptotics with a \(y\)-wise discrete behaviour.Metastable dynamics for a hyperbolic variant of the mass conserving Allen-Cahn equation in one space dimension.https://www.zbmath.org/1456.350112021-04-16T16:22:00+00:00"Folino, Raffaele"https://www.zbmath.org/authors/?q=ai:folino.raffaeleSummary: In this paper, we consider some hyperbolic variants of the mass conserving Allen-Cahn equation, which is a nonlocal reaction-diffusion equation, introduced (as a simpler alternative to the Cahn-Hilliard equation) to describe phase separation in binary mixtures. In particular, we focus our attention on the metastable dynamics of solutions to the equation in a bounded interval of the real line with homogeneous Neumann boundary conditions. It is shown that the evolution of profiles with \(N + 1\) transition layers is very slow in time and we derive a system of ODEs, which describes the exponentially slow motion of the layers. A comparison with the classical Allen-Cahn and Cahn-Hilliard equations and theirs hyperbolic variations is also performed.Global existence and boundedness of a forager-exploiter system with nonlinear diffusions.https://www.zbmath.org/1456.352032021-04-16T16:22:00+00:00"Wang, Jianping"https://www.zbmath.org/authors/?q=ai:wang.jianping.1|wang.jianpingA system of three parabolic equations combining chemotaxis and reaction terms is studied in bounded domains of \(\mathbb R^n\), under the homogeneous Neumann conditions. This is, the so-called, forager-exploiter (a.k.a. producer-scrounger) model. The main result of the paper is the global-in-time existence of solutions (and their boundedness) when the diffusion coefficients are nonlinear (and nondegenerate).
Reviewer: Piotr Biler (Wrocław)Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term.https://www.zbmath.org/1456.350802021-04-16T16:22:00+00:00"Giga, Yoshikazu"https://www.zbmath.org/authors/?q=ai:giga.yoshikazu"Požár, Norbert"https://www.zbmath.org/authors/?q=ai:pozar.norbertSummary: A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system.https://www.zbmath.org/1456.351302021-04-16T16:22:00+00:00"Muatjetjeja, Ben"https://www.zbmath.org/authors/?q=ai:muatjetjeja.ben"Mothibi, Dimpho Millicent"https://www.zbmath.org/authors/?q=ai:mothibi.dimpho-millicent"Khalique, Chaudry Masood"https://www.zbmath.org/authors/?q=ai:khalique.chaudry-masoodSummary: In this paper we perform Lie group classification of a generalized coupled (2+1)-dimensional hyperbolic system, viz., \( u_{tt}-u_{xx}-u_{yy}+f(v) = 0,\,v_{tt}-v_{xx}-v_{yy}+g(u) = 0 \), which models many physical phenomena in nonlinear sciences. We show that the Lie group classification of the system provides us with an eleven-dimensional equivalence Lie algebra, whereas the principal Lie algebra is six-dimensional and has several possible extensions. It is further shown that several cases arise in classifying the arbitrary functions \( f \) and \( g \), the forms of which include, amongst others, the power and exponential functions. Finally, for three cases we carry out symmetry reductions for the coupled system.A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation.https://www.zbmath.org/1456.350872021-04-16T16:22:00+00:00"Khalique, Chaudry Masood"https://www.zbmath.org/authors/?q=ai:khalique.chaudry-masood"Moleleki, Letlhogonolo Daddy"https://www.zbmath.org/authors/?q=ai:moleleki.letlhogonolo-daddySummary: This paper aims to study a generalized first extended (3+1)-dimensional Jimbo-Miwa equation. Symmetry reductions on this equation are performed several times and it is reduced to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is found in terms of the incomplete elliptic integral function. Also exact solutions are constructed using the \(({G'}/{G})\)-expansion method. Thereafter the conservation laws of the underlying equation are computed by invoking the conservation theorem due to Ibragimov. The conservation laws obtained contain an energy conservation law and three momentum conservation laws.Decay property for symmetric hyperbolic system with memory-type diffusion.https://www.zbmath.org/1456.350372021-04-16T16:22:00+00:00"Okada, Mari"https://www.zbmath.org/authors/?q=ai:okada.mari"Mori, Naofumi"https://www.zbmath.org/authors/?q=ai:mori.naofumi"Kawashima, Shuichi"https://www.zbmath.org/authors/?q=ai:kawashima.shuichiSummary: We study the decay property for symmetric hyperbolic systems with memory-type diffusion. Under the structural condition (called Craftsmanship condition) we prove that the system is uniformly dissipative and the solutions satisfy the corresponding decay property. Our proof is based on a technical energy method in the Fourier space which makes use of the properties of strongly positive definite kernels.Schrödinger heat kernel upper bounds on gradient shrinking Ricci solitons.https://www.zbmath.org/1456.350992021-04-16T16:22:00+00:00"Wu, Jia-Yong"https://www.zbmath.org/authors/?q=ai:wu.jiayongSummary: In this paper we give new Gaussian type upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons with the scalar curvature bounded above. This result is a little broader than our earlier paper at some cases. The proof uses on a Davies type integral estimate and a local mean value inequality on gradient shrinking Ricci solitons.Some singular equations modeling MEMS.https://www.zbmath.org/1456.351932021-04-16T16:22:00+00:00"Laurençot, Philippe"https://www.zbmath.org/authors/?q=ai:laurencot.philippe"Walker, Christoph"https://www.zbmath.org/authors/?q=ai:walker.christophThis paper can be seen as a review about the derivation and the known (and unknown) mathematical results associated to the two dimensional version of microelectromechanical systems. The systems of equations describing these problems are determined by different models and parameters. Results about locally well-posedness, global solutions and stable stationary solutions are recalled depending on the values of these parameters and the precise models. The authors give a good description of the known (and unkown) results in a table where local and global existence and finite time singularity for the evolution problem as well as about existence of steady states are recalled.
Reviewer: Ramón Quintanilla De Latorre (Barcelona)Partial symmetry of normalized solutions for a doubly coupled Schrödinger system.https://www.zbmath.org/1456.350082021-04-16T16:22:00+00:00"Luo, Haijun"https://www.zbmath.org/authors/?q=ai:luo.haijun"Zhang, Zhitao"https://www.zbmath.org/authors/?q=ai:zhang.zhitaoSummary: We consider the normalized solutions of a Schrödinger system which arises naturally from nonlinear optics, the Hartree-Fock theory for Bose-Einstein condensates. And we investigate the partial symmetry of normalized solutions to the system and their symmetry-breaking phenomena. More precisely, when the underlying domain is bounded and radially symmetric, we develop a kind of polarization inequality with weight to show that the first two components of the normalized solutions are foliated Schwarz symmetric with respect to the same point, while the latter two components are foliated Schwarz symmetric with respect to the antipodal point. Furthermore, by analyzing the singularly perturbed limit profiles of these normalized solutions, we prove that they are not radially symmetric at least for large nonlinear coupling constant \(\beta \), which seems a new method to prove the symmetry-breaking phenomenons of normalized solutions.Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations.https://www.zbmath.org/1456.350392021-04-16T16:22:00+00:00"Poláčik, P."https://www.zbmath.org/authors/?q=ai:polacik.peterSummary: We consider bounded solutions of the semilinear heat equation \(u_t=u_{xx}+f(u)\) on \(R\), where \(f\) is of the unbalanced bistable type. We examine the \(\omega\)-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at \(x=\pm\infty\), the \(\omega\)-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of \(f\).On a partial \(q\)-analog of a singularly perturbed problem with Fuchsian and irregular time singularities.https://www.zbmath.org/1456.350852021-04-16T16:22:00+00:00"Malek, Stephane"https://www.zbmath.org/authors/?q=ai:malek.stephaneSummary: A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs \(q\)-Gevrey and Gevrey bounds.Hyers-Ulam stability of a nonautonomous semilinear equation with fractional diffusion.https://www.zbmath.org/1456.352242021-04-16T16:22:00+00:00"Villa-Morales, José"https://www.zbmath.org/authors/?q=ai:villa-morales.joseSummary: In this paper, we study the Hyers-Ulam stability of a nonautonomous semilinear reaction-diffusion equation. More precisely, we consider a nonautonomous parabolic equation with a diffusion given by the fractional Laplacian. We see that such a stability is a consequence of a Gronwall-type inequality.Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues.https://www.zbmath.org/1456.351342021-04-16T16:22:00+00:00"Banach, Zbigniew"https://www.zbmath.org/authors/?q=ai:banach.zbigniew"Larecki, Wieslaw"https://www.zbmath.org/authors/?q=ai:larecki.wieslaw"Ruggeri, Tommaso"https://www.zbmath.org/authors/?q=ai:ruggeri.tommasoSummary: The results of a previous paper [\textit{A. Muracchini} et al., ibid. 15, No. 2, 143--158 (1992; Zbl 0775.35004)] are generalized by considering a hyperbolic system in one space dimension with multiple eigenvalues. The dispersion relation for linear plane waves in the high-frequency limit is analyzed and the recurrence formulas for the phase velocity and the attenuation factor are derived in terms of the coefficients of a formal series expansion in powers of the reciprocal of frequency. In the case of multiple eigenvalues, it is also verified that linear stability implies \(\lambda\)-stability for the waves of weak discontinuity. Moreover, for the linearized system, the relationship between entropy and stability is studied. When the nonzero eigenvalue is simple, the results of the paper mentioned above are recovered. In order to illustrate the procedure, an example of the linear hyperbolic system is presented in which, depending on the values of parameters, the multiplicity of nonzero eigenvalues is either one or two. This example describes the dynamics of a mixture of two interacting phonon gases.Layer potentials for Lamé systems and homogenization of perforated elastic medium with clamped holes.https://www.zbmath.org/1456.350202021-04-16T16:22:00+00:00"Jing, Wenjia"https://www.zbmath.org/authors/?q=ai:jing.wenjiaThe author obtains quantitative homogenization results for a (perforated) domain clamped at the boundary of the holes. The employed method is based on layer potentials. For a suitable scaling of the holes (etc.), one also proves corrector estimates.
Reviewer: Adrian Muntean (Karlstad)Lie symmetry analysis, traveling wave solutions, and conservation laws to the \((3+1)\)-dimensional generalized B-type Kadomtsev-Petviashvili equation.https://www.zbmath.org/1456.350712021-04-16T16:22:00+00:00"Yang, Huizhang"https://www.zbmath.org/authors/?q=ai:yang.huizhang"Liu, Wei"https://www.zbmath.org/authors/?q=ai:liu.wei.6|liu.wei|liu.wei.5|liu.wei.3|liu.wei.1|liu.wei.7|liu.wei.8|liu.wei.9"Zhao, Yunmei"https://www.zbmath.org/authors/?q=ai:zhao.yunmeiSummary: In this paper, the \((3+1)\)-dimensional generalized B-type Kadomtsev-Petviashvili(BKP) equation is studied applying Lie symmetry analysis. We apply the Lie symmetry method to the \((3+1)\)-dimensional generalized BKP equation and derive its symmetry reductions. Based on these symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method. Finally, the conservation laws to the \((3+1)\)-dimensional generalized BKP equation are presented by invoking the multiplier method.Partial regularity for the steady hyperdissipative fractional Navier-Stokes equations.https://www.zbmath.org/1456.350582021-04-16T16:22:00+00:00"Chen, Eric"https://www.zbmath.org/authors/?q=ai:chen.eric-bingshu|chen.eric-zhi|chen.eric-ySummary: We extend the Caffarelli-Kohn-Nirenberg type partial regularity theory for the steady 5-dimensional fractional Navier-Stokes equations with external force to the hyperdissipative setting. In our argument we use the methods of Colombo-De Lellis-Massaccesi to apply a blowup procedure adapted from work of Ladyzhenskaya-Seregin.Interior estimates in the sup-norm for a class of generalized functions with integral representations.https://www.zbmath.org/1456.350512021-04-16T16:22:00+00:00"Ariza, Eusebio"https://www.zbmath.org/authors/?q=ai:ariza.eusebio"Di Teodoro, Antonio"https://www.zbmath.org/authors/?q=ai:di-teodoro.antonio-nicola"Vanegas, Judith"https://www.zbmath.org/authors/?q=ai:vanegas.judith-cSummary: In this paper we construct apriori estimates for the first order derivatives in the sup-norm for first order meta-monogenic functions, generalized monogenic functions satisfying a differential equation with an anti-monogenic right hand side and generalized meta-monogenic functions satisfying a differential equation with an anti-meta-monogenic right hand side. We obtain such estimates through integral representations of these classes of functions and give an explicit expression for the corresponding constants appearing in the estimates. Then we show how initial value problems can be solved in case an interior estimate is true in the function spaces under consideration. All related functions are in a Clifford type algebra.Unfolding homogenization in doubly periodic media and applications.https://www.zbmath.org/1456.350162021-04-16T16:22:00+00:00"Bunoiu, Renata"https://www.zbmath.org/authors/?q=ai:bunoiu.renata"Donato, Patrizia"https://www.zbmath.org/authors/?q=ai:donato.patriziaSummary: We define a reiterated unfolding operator for a doubly periodic domain presenting two periodicity scales. Then we show how to apply it to the homogenization of both linear and nonlinear problems. The main novelty is that this method allows the use of test functions with one scale of periodicity only and it considerably simplifies the proofs of the convergence results. We illustrate this new approach on a Poisson problem with Dirichlet boundary conditions and on the flow of a power law fluid in a doubly periodic porous medium.Remarks on the nonlocal Dirichlet problem.https://www.zbmath.org/1456.350592021-04-16T16:22:00+00:00"Grzywny, Tomasz"https://www.zbmath.org/authors/?q=ai:grzywny.tomasz"Kassmann, Moritz"https://www.zbmath.org/authors/?q=ai:kassmann.moritz"Leżaj, Łukasz"https://www.zbmath.org/authors/?q=ai:lezaj.lukaszSummary: We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data.https://www.zbmath.org/1456.351232021-04-16T16:22:00+00:00"Dal Maso, Gianni"https://www.zbmath.org/authors/?q=ai:dal-maso.gianni"Lucardesi, Ilaria"https://www.zbmath.org/authors/?q=ai:lucardesi.ilariaSummary: Given a bounded open set \(\Omega \subset \mathbb{R}^d\) with Lipschitz boundary and an increasing family \(\Gamma_t\), \(t \in [0, T]\), of closed subsets of \(\Omega\), we analyze the scalar wave equation \(\ddot{u} - \operatorname{div}(A \nabla u) = f\) in the time varying cracked domains \(\Omega \backslash \Gamma_t\). Here we assume that the sets \(\Gamma_t\) are contained into a \textit{prescribed} \((d - 1)\)-manifold of class \(C^{2}\). Our approach relies on a change of variables: recasting the problem on the reference configuration \(\Omega \backslash \Gamma_0\), we are led to consider a hyperbolic problem of the form \(\ddot{v} - \operatorname{div}(B \nabla v)+a\cdot\nabla v -2b\cdot\nabla\dot{v} = g\) in \(\Omega \backslash \Gamma_0\). Under suitable assumptions on the regularity of the change of variables that transforms \(\Omega \backslash \Gamma_t\) into \(\Omega \backslash \Gamma_0\), we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by-product, the continuous dependence of the solutions with respect to the cracks.A note on the blow-up criterion for the compressible isentropic Navier-Stokes equations with vacuum.https://www.zbmath.org/1456.761092021-04-16T16:22:00+00:00"Liu, Shengquan"https://www.zbmath.org/authors/?q=ai:liu.shengquan"Zhang, Jianwen"https://www.zbmath.org/authors/?q=ai:zhang.jianwen(no abstract)Estimate for evolutionary surfaces of prescribed mean curvature and the convergence.https://www.zbmath.org/1456.350542021-04-16T16:22:00+00:00"Wang, Peihe"https://www.zbmath.org/authors/?q=ai:wang.peihe"Gao, Xinyu"https://www.zbmath.org/authors/?q=ai:gao.xinyuSummary: In the paper, we will discuss the gradient estimate for the evolutionary surfaces of prescribed mean curvature with Neumann boundary value under the condition \(f_\tau\ge -\kappa \), which is the same as the one in the interior estimate by K. Ecker and generalizes the condition \(f_\tau\ge 0\) studied by Gerhardt etc. Also, based on the elliptic result obtained recently, we will show the longtime behavior of surfaces moving by the velocity being equal to the mean curvature.Bifurcations of traveling wave solutions for a generalized Camassa-Holm equation.https://www.zbmath.org/1456.350752021-04-16T16:22:00+00:00"Wei, Minzhi"https://www.zbmath.org/authors/?q=ai:wei.minzhi"Sun, Xianbo"https://www.zbmath.org/authors/?q=ai:sun.xianbo"Zhu, Hongying"https://www.zbmath.org/authors/?q=ai:zhu.hongyingSummary: In this paper, the traveling wave solutions for a generalized Camassa-Holm equation \(u_t-u_{xxt}=\frac{1}{2}(p+1)(p+2)u^pu_x-\frac{1}{2}p(p-1)u^{p-2}u_x^3-2pu^{p-1}u_xu_{xx}-u^pu_{xxx}\) are investigated. By using the bifurcation method of dynamical systems, three major results for this equation are highlighted. First, there are one or two singular straight lines in the two-dimensional system under some different conditions. Second, all the bifurcations of the generalized Camassa-Holm equation are given for \(p\) either positive or negative integer. Third, we prove that the corresponding traveling wave system of this equation possesses peakon, smooth solitary wave solution, kink and anti-kink wave solution, and periodic wave solutions.A fully-mixed formulation for the steady double-diffusive convection system based upon Brinkman-Forchheimer equations.https://www.zbmath.org/1456.651552021-04-16T16:22:00+00:00"Caucao, Sergio"https://www.zbmath.org/authors/?q=ai:caucao.sergio"Gatica, Gabriel N."https://www.zbmath.org/authors/?q=ai:gatica.gabriel-n"Oyarzúa, Ricardo"https://www.zbmath.org/authors/?q=ai:oyarzua.ricardo"Sánchez, Nestor"https://www.zbmath.org/authors/?q=ai:sanchez.nestor-eSummary: We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman-Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman-Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in \(\text{L}^4\). As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable \(\text{H}(\operatorname{div})\)-type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax-Milgram and Banach-Nečas-Babuška theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart-Thomas spaces of order \(k\geq 0\) for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree \(\leq k+1\) are employed for the velocity, and piecewise polynomials of degree \(\leq k\) for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.Time-fractional Allen-Cahn equations: analysis and numerical methods.https://www.zbmath.org/1456.650622021-04-16T16:22:00+00:00"Du, Qiang"https://www.zbmath.org/authors/?q=ai:du.qiang"Yang, Jiang"https://www.zbmath.org/authors/?q=ai:yang.jiang"Zhou, Zhi"https://www.zbmath.org/authors/?q=ai:zhou.zhiSummary: In this work, we consider a time-fractional Allen-Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order \(\alpha \in (0,1)\). First, the well-posedness and (limited) smoothing property are studied, by using the maximal \(L^p\) regularity of fractional evolution equations and the fractional Grönwall's inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall's inequality and maximal \(\ell^p\) regularity, we prove that the convergence rates of those time-stepping schemes are \(O(\tau^\alpha)\) without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen-Cahn dynamics.Generic regularity of free boundaries for the obstacle problem.https://www.zbmath.org/1456.352342021-04-16T16:22:00+00:00"Figalli, Alessio"https://www.zbmath.org/authors/?q=ai:figalli.alessio"Ros-Oton, Xavier"https://www.zbmath.org/authors/?q=ai:ros-oton.xavier"Serra, Joaquim"https://www.zbmath.org/authors/?q=ai:serra.joaquimIn this very interesting paper the authors establish the generic regularity of free boundaries for the obstacle problem in \(\mathbb{R}^n\). Classical results of Caffarelli guarantee that the free boundary is \(C^{\infty}\) outside of a set of singular points, but examples show that such set could be in general \((n-1)\)-dimensional. In this paper, the authors show that, generically, the singular set has zero \(\mathcal{H}^{n-4}\) measure. Therefore, for \(n\le 4\), the free boundary is generically a \(C^{\infty}\) manifold, solving a conjecture of Schaeffer. Such conjecture stated that generically, free boundaries in the obstacle problem have no singular points.
To describe the results proved in more detail, consider the obstacle problem
\[
\Delta u= \chi_{\{u>0\}}, \ \ \ u\ge 0 \ \ \text{ in } \Omega\subset\mathbb{R}^n.
\]
The authors are able to prove Schaeffer's conjecture in \(\mathbb{R}^3\) and \(\mathbb{R}^4\). To do so, they consider a monotone family of solutions \(\{u^t\}_{t\in (-1,1)}\) to the obstacle problem in \(B_1\) satisfying a ``uniform monotonicity'' condition. Such condition rules out the existence of regions that remain stationary as \(t\) increases. Assuming \((-1,1)\ni t\mapsto u^t\Big|_{\partial B_1}\in L^{\infty}(\partial B_1)\) is continuous with respect to \(t\), the authors prove that \(\mathcal{H}^{n-4}(\Sigma^t)=0\) for a.e. \(t\in (-1,1)\). Here \(\Sigma^t\) is the set of singular points for \(u^t\). This, in particular, establishes Schaeffer's conjecture for \(n\le 4\).
The authors also prove an analogous result to solutions to the Hele-Shaw flow.
A key ingredient in the proof of their main result is a fine understanding of the singular points. In particular, the authors establish a very interesting new higher order expansion at most singular points for monotone families of solutions to the obstacle problem.
Reviewer: Mariana Vega Smit (Bellingham)Quasi-periodic solutions of PDEs.https://www.zbmath.org/1456.350102021-04-16T16:22:00+00:00"Berti, Massimiliano"https://www.zbmath.org/authors/?q=ai:berti.massimilianoSummary: The aim of this talk is to present some recent existence results about quasi-periodic solutions for PDEs like nonlinear wave and Schrödinger equations in \(\mathbb{T}^d\),\(d\geq 2\), and the 1-d derivative wave equation. The proofs are based on both Nash-Moser implicit function theorems and KAM theory.Decay of solutions for 2D Navier-Stokes equations posed on Lipschitz and smooth bounded and unbounded domains.https://www.zbmath.org/1456.351532021-04-16T16:22:00+00:00"Larkin, N. A."https://www.zbmath.org/authors/?q=ai:larkin.nickolai-a|larkin.nikolai-a|larkin.nikolaj-a"Padilha, M. V."https://www.zbmath.org/authors/?q=ai:padilha.m-vThe present paper deals with the four initial-boundary value problems for the 2D Navier-Stokes equations posed on a rectangle, on a half-strip, on a bounded smooth domain, on an unbounded smooth domain. The existence and uniqueness of strong solutions are established. It is proved that strong solutions in unbounded domains are regular in smooth bounded subdomains, the decay rate is different for different norms, depending on the geometrical characteristics of the domains. The existence of an unique global regular solution, which decays exponentially as \(t\rightarrow +\infty\), is proved. Sharp estimates for the exponential decay rates of solutions to initial-boundary value problems for the 2D Navier-Stocks equations are determined.
Reviewer: Georg V. Jaiani (Tbilisi)Exact solutions and conservation laws of multi Kaup-Boussinesq system with fractional order.https://www.zbmath.org/1456.352212021-04-16T16:22:00+00:00"Singla, Komal"https://www.zbmath.org/authors/?q=ai:singla.komal"Rana, M."https://www.zbmath.org/authors/?q=ai:rana.mehwish|rana.meenakshiSummary: The purpose of the present work is to investigate exact solutions of the fractional order multi Kaup-Boussinesq system with \(l=2\) by using the group invariance approach and power series expansion method. Due to the significance of conserved vectors in terms of integrability and behaviour of nonlinear systems, the conservation laws are also derived by testing the nonlinear self-adjointness.Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains.https://www.zbmath.org/1456.351492021-04-16T16:22:00+00:00"Nazarov, Sergei Aleksandrovich"https://www.zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: Full asymptotic expansions are constructed and justified for two series of eigenvalues and the corresponding eigenfunctions of the spectral Steklov problem in a domain with a singular boundary perturbation having the form of a small cavity. The terms of those series are of type \( \lambda _k+o(1)\) and \( \varepsilon ^{-1}(\mu _m+o(1))\), where \( \lambda _k\) and \( \mu _m\) are the eigenvalues of the Steklov problem in a bounded domain without cavity and the exterior Steklov problem for a cavity of unit size. A similar problem of the surface wave is also treated. The smoothness requirements on the boundary are discussed and unsolved problems are stated.Some models for the interaction of long and short waves in dispersive media. I: Derivation.https://www.zbmath.org/1456.351572021-04-16T16:22:00+00:00"Nguyen, Nghiem V."https://www.zbmath.org/authors/?q=ai:nguyen.nghiem-v"Liu, Chuangye"https://www.zbmath.org/authors/?q=ai:liu.chuangyeSummary: It is universally accepted that the cubic, nonlinear Schrödinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Korteweg-de Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves, and such a system has been studied over the last few decades. However, questions about the validity of this system in the study of water waves were raised in a previous work of one of us where the analysis was presented using the fifth-order KdV as the starting point. These questions will now be settled unequivocally in a series of papers. In this first part, we show that the NLS-KdV system (or even the \textit{linear} Schrödinger-KdV system) cannot be resulted from the full Euler equations formulated in the study of water waves. In the process of so doing, we also propose a few alternative models for describing the interaction of long and short waves.Regularity theory of elliptic systems in \(\varepsilon\)-scale flat domains.https://www.zbmath.org/1456.350242021-04-16T16:22:00+00:00"Zhuge, Jinping"https://www.zbmath.org/authors/?q=ai:zhuge.jinpingThe paper under review deals with the boundary regularity of elliptic systems and equations in a bounded domain with arbitrarily rough boundary at small scales. The author considers an equation of the type
\[
\nabla \cdot \Big(A(x/\varepsilon) \nabla u_\varepsilon(x)\Big)=0\,
\]
in rough domains and obtains large-scale Lipschitz estimate without any regularity assumption, except for a quantitative large-scale flatness assumption. The result provides a mathematical explanation for the fact that boundary regularity of the solutions of PDEs should be physically and experimentally expected even if the surface of the medium is arbitrarily rough at small scale.
Reviewer: Paolo Musolino (Padova)The geometry of synchronization problems and learning group actions.https://www.zbmath.org/1456.051052021-04-16T16:22:00+00:00"Gao, Tingran"https://www.zbmath.org/authors/?q=ai:gao.tingran"Brodzki, Jacek"https://www.zbmath.org/authors/?q=ai:brodzki.jacek"Mukherjee, Sayan"https://www.zbmath.org/authors/?q=ai:mukherjee.sayanSummary: We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group \(G\) on connected graph \(\Gamma\) with a flat principal \(G\)-bundle over \(\Gamma\), thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of \(\Gamma\) into \(G\). We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal \(G\)-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham-Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions -- partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations -- and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.Sharp blow up estimates and precise asymptotic behavior of singular positive solutions to fractional Hardy-Hénon equations.https://www.zbmath.org/1456.352272021-04-16T16:22:00+00:00"Yang, Hui"https://www.zbmath.org/authors/?q=ai:yang.hui.1"Zou, Wenming"https://www.zbmath.org/authors/?q=ai:zou.wenmingSummary: In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation
\[
(-\Delta)^\sigma u = |x|^\alpha u^p \qquad \text{in } B_1 \backslash \{0\}
\]
with an isolated singularity at the origin, where \(\sigma \in (0, 1)\) and the punctured unit ball \(B_1 \backslash \{0\} \subset \mathbb{R}^n\) with \(n \geq 2\). When \(-2 \sigma < \alpha < 2 \sigma\) and \(\frac{n + \alpha}{n - 2 \sigma} < p < \frac{n + 2 \sigma}{n - 2 \sigma}\), we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by \textit{B. Gidas} and \textit{J. Spruck} [Commun. Pure Appl. Math. 34, 525--598 (1981; Zbl 0465.35003)], but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on \(\mathbb{S}_+^n\). We also investigate isolated singularities located at infinity of fractional Hardy-Hénon equations.Asymptotic dynamics for reaction diffusion equations in unbounded domain.https://www.zbmath.org/1456.350412021-04-16T16:22:00+00:00"Li, Yongjun"https://www.zbmath.org/authors/?q=ai:li.yongjun"Wei, Jinying"https://www.zbmath.org/authors/?q=ai:wei.jinyingSummary: In this paper we study the asymptotic dynamics for reaction diffusion equation defined in \(\mathbb{R}^n\). We will prove that the equation possesses a fixed point when the nonlinearity satisfies some restrictive conditions and then we show that the fixed point is an exponential attractor.Large time behavior, bi-Hamiltonian structure, and kinetic formulation for a complex Burgers equation.https://www.zbmath.org/1456.350332021-04-16T16:22:00+00:00"Gao, Yu"https://www.zbmath.org/authors/?q=ai:gao.yu"Gao, Yuan"https://www.zbmath.org/authors/?q=ai:gao.yuan"Liu, Jian-Guo"https://www.zbmath.org/authors/?q=ai:liu.jian-guoSummary: We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex plane. These solutions converge to a steady state given by Wigner's semicircle law. A unique global weak solution with nonnegative initial data to the Dyson equation is obtained, and some explicit solutions are given by Wigner's semicircle laws. We also construct a bi-Hamiltonian structure for the system of real and imaginary components of the complex Burgers equation (coupled Burgers system). We establish a kinetic formulation for the coupled Burgers system and prove the existence and uniqueness of entropy solutions. The coupled Burgers system in Lagrangian variable naturally leads to two interacting particle systems, the Fermi-Pasta-Ulam-Tsingou model with nearest-neighbor interactions, and the Calogero-Moser model. These two particle systems yield the same Lagrangian dynamics in the continuum limit.Global higher integrability of solutions to subelliptic double obstacle problems.https://www.zbmath.org/1456.350932021-04-16T16:22:00+00:00"Du, Guangwei"https://www.zbmath.org/authors/?q=ai:du.guangwei"Li, Fushan"https://www.zbmath.org/authors/?q=ai:li.fushanSummary: In this paper we consider the double obstacle problems associated with nonlinear subelliptic equation
\[X^*A(x,u,Xu)+ B(x,u,Xu)=0, \quad x\in\Omega,\]
where \(X=(X_1,\ldots,X_m)\) is a system of smooth vector fields defined in \(\mathbb{R}^n\) satisfying Hörmander's condition. The global higher integrability for the gradients of the solutions is obtained under a capacitary assumption on the complement of the domain \(\Omega \).Oscillatory behavior of a fractional partial differential equation.https://www.zbmath.org/1456.352262021-04-16T16:22:00+00:00"Wang, Jiangfeng"https://www.zbmath.org/authors/?q=ai:wang.jiangfeng"Meng, Fanwei"https://www.zbmath.org/authors/?q=ai:meng.fanweiSummary: In this paper, a fractional partial differential equation subject to the Robin boundary condition is considered. Based on the properties of Riemann-Liouville fractional derivative and a generalized Riccati technique, we obtained sufficient conditions for oscillation of the solutions of such equation. Examples are given to illustrate the main results.Local existence for an isentropic compressible Navier-Stokes-\(P1\) approximate model arising in radiation hydrodynamics.https://www.zbmath.org/1456.351522021-04-16T16:22:00+00:00"Fan, Jishan"https://www.zbmath.org/authors/?q=ai:fan.jishan"Hu, Yuxi"https://www.zbmath.org/authors/?q=ai:hu.yuxi"Nakamura, Gen"https://www.zbmath.org/authors/?q=ai:nakamura.genSummary: In this paper we prove the local existence of strong solutions to an isentropic compressible Navier-Stokes-\(P1\) approximate model arising in radiation hydrodynamics in a bounded domain with vacuum.Determining nodes of the global attractor for an incompressible non-Newtonian fluid.https://www.zbmath.org/1456.350442021-04-16T16:22:00+00:00"Zhao, Caidi"https://www.zbmath.org/authors/?q=ai:zhao.caidi"Li, Yanjiao"https://www.zbmath.org/authors/?q=ai:li.yanjiao"Zhang, Mingshu"https://www.zbmath.org/authors/?q=ai:zhang.mingshuSummary: This paper estimates the finite number of the determining nodes to the equations for an incompressible non-Newtonian fluid with space-periodic or no-slip boundary conditions. The authors prove that, whenever the second order derivatives of two different solutions within the global attractor have the same time-asymptotic behavior at finite number of points in the physical space, then the two solutions possess the same time-asymptotic behavior at almost everywhere points of the physical space.The Stokes and Navier-Stokes equations in exterior domains. Moving domains and decay properties.https://www.zbmath.org/1456.350022021-04-16T16:22:00+00:00"Wegmann, David"https://www.zbmath.org/authors/?q=ai:wegmann.davidIn der vorliegenden Arbeit beschäftigt sich der Autor mit zwei verschiedenen Problemstellungen zu den Navier-Stokes-Gleichungen
\begin{align*}
u_t-\nu\Delta u+u\cdot\nabla u+\nabla p&=f \qquad \text{in }\Omega\times(0,T),\\
\text{div}\,u&=0 \qquad \text{in } \Omega\times(0,T),\\
u&=\beta \qquad \text{auf }\partial\Omega\times (0,T), \\
u(0)&=u_0 \qquad \text{auf }\Omega.
\end{align*}
Diese sind das heutzutage meist anerkannte System zur Modellierung der Fließgeschwindigkeit inkompressibler, Newton'scher Flüssigkeiten wie z.B. Wasser.
Betrachtet man nun ein Objekt mit sehr hoher Dichte, dass in einem mit Wasser gefüllten Behälter aufgrund einer äußeren Kraft, z.B. wegen der Gravitation, sinkt, so verändert sich das mit Wasser gefüllte Gebiet. Aufgrund der hohen Dichte des sich bewegenden Objekts ist es in erster Näherung sinnvoll, die Navier-Stokes-Gleichungen in einem nicht zylindrischen Raum-Zeit-Gebiet zu betrachten. Eine anschließende Transformation auf ein zylindrisches Raum-Zeit-Gebiet führt zu einem nicht-autonomen System, das im ersten Abschnitt dieser Arbeit untersucht wird. Hierbei betrachtet der Autor den Fall, dass zu jedem Zeitpunkt das räumliche Gebiet ein Außenraumgebiet sei.
In Kapitel 2 wird bewiesen, dass das zugehörige linearisierte System Maximale Regularität in \(L^s(0,\infty; L^q_\sigma)\) besitzt, falls \(1<q<\frac 32\) und \(1<s<\infty\).
Basierend auf diesem Resultat, wird in Kapitel 3 das nichtlineare System behandelt und gezeigt, dass dieses für beliebig große Daten eine zeitlokale Lösung und für kleine Daten eine zeitglobale Lösung besitzt. Die Argumentation in diesem Kapitel basiert im Wesentlichen auf der Verwendung des Banach'schen Fixpunktsatzes und benötigt eine geeignete Abschätzung des nichtlinearem Terms.
In den beiden finalen Kapiteln dieser Arbeit wird das Navier-Stokes-System in einem zylindrischen Raum-Zeit-Gebiet mit inhomogenen Dirichlet Randdaten betrachtet. Üblicherweise und so auch in dieser Arbeit, wird eine schwache Lösung \(u\) der Navier-Stokes-Gleichungen mit inhomogenen Randdaten konstruiert als \(u=v+b\), wobei \(b\) eine Fortsetzung der Randdaten und \(v\) eine schwache Lösung zu einer zur Navier-Stokes-Gleichung ähnlichen Differenzialgleichung gegeben durch
\[\begin{aligned}
v_t-\Delta u+(v+b)\cdot\nabla(v+b)+\nabla p&=f\qquad \text{in }\Omega \times(0,T),\\
\text{div}\,v&=0\qquad \text{in }\Omega\times(0,T),\\
v&=0\qquad \text{auf }\partial\Omega \times(0,T),\\
v(0)&=u_0\qquad \text{auf }\Omega
\end{aligned}\tag{PNST}\]
bezeichnet. Es wird bewiesen, dass eine schwache Lösung \(v\) zu (PNST) existiert, die exponentiell abklingt, wenn \(\Omega\) beschränkt ist, und wie \(t^{-\frac 34+\varepsilon}\) für ein beliebiges \(\varepsilon >0\) unter geeigneten Annahmen an die Daten, falls \(\Omega\) ein Außenraumgebiet ist. Der Beweis dieser Aussage basiert wesentlich auf dem Konstruktionsverfahren der schwachen Lösung, somit kann zunächst nur eine Aussage zur Existenz einer abklingenden Lösung getroffen werden.
Durch den Beweis einer Verallgemeinerung des Eindeutigkeitssatzes von Serrin wird abschließend gezeigt, dass die Abklingeigenschaft einer speziell konstruierten Lösung sich unter geeigneten Voraussetzungen auf alle schwachen Lösungen, die der starken Energieungleichung genügen, übertragen lässt.
Die Arbeit ist anspruchsvoll und enthält durchaus tiefliegende Ergebnisse. Leider enthält sie so gut wie keine Beispiele, was das Lesen ziemlich trocken und langweilig gestaltet.
Reviewer: Jürgen Appell (Würzburg)On the stability of a population model with nonlocal dispersal.https://www.zbmath.org/1456.350402021-04-16T16:22:00+00:00"Sun, Jianwen"https://www.zbmath.org/authors/?q=ai:sun.jianwen"Wang, Chong"https://www.zbmath.org/authors/?q=ai:wang.chong|wang.chong.4|wang.chong.7|wang.chong.6Summary: This paper is concerned with a nonlocal dispersal population model with spatial competition and aggregation. We establish the existence and uniqueness of positive solutions by the method of coupled upper-lower solutions. We obtain the global stability of the stationary solutions.Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria.https://www.zbmath.org/1456.350382021-04-16T16:22:00+00:00"Pierre, Michel"https://www.zbmath.org/authors/?q=ai:pierre.michel"Suzuki, Takashi"https://www.zbmath.org/authors/?q=ai:suzuki.takashi"Umakoshi, Haruki"https://www.zbmath.org/authors/?q=ai:umakoshi.harukiSummary: We consider the asymptotic behavior for large time of solutions to reaction-diffusion systems modeling reversible chemical reactions. We focus on the case where multiple equilibria exist. In this case, due to the existence of so-called ``boundary equilibria'', the analysis of the asymptotic behavior is not obvious. The solution is understood in a weak sense as a limit of adequate approximate solutions. We prove that this solution converges in \(L^1\) toward an equilibrium as time goes to infinity and that the convergence is exponential if the limit is strictly positive.Invariant measures for systems of Kolmogorov equations.https://www.zbmath.org/1456.350312021-04-16T16:22:00+00:00"Addona, Davide"https://www.zbmath.org/authors/?q=ai:addona.davide"Angiuli, Luciana"https://www.zbmath.org/authors/?q=ai:angiuli.luciana"Lorenzi, Luca"https://www.zbmath.org/authors/?q=ai:lorenzi.lucaSummary: In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity.A note on the linear stability of the steady state of a nonlinear renewal equation with a parameter.https://www.zbmath.org/1456.350272021-04-16T16:22:00+00:00"Tumuluri, Suman Kumar"https://www.zbmath.org/authors/?q=ai:tumuluri.suman-kumarSummary: In this article we consider a variant of age-structured nonlinear Lebowitz-Rubinow equation. We study the linear stability of this equation near the nontrivial steady state by analyzing the corresponding characteristic equation. In particular, we provide some sufficient conditions under which the nonzero steady state is linearly stable.Global convergence of an isentropic Euler-Poisson system in \(\mathbb{R}^+ \times \mathbb{R}^d\).https://www.zbmath.org/1456.350342021-04-16T16:22:00+00:00"Huimin, Tian Yue-Jun Peng"https://www.zbmath.org/authors/?q=ai:huimin.tian-yue-jun-peng"Lingling, Zhang"https://www.zbmath.org/authors/?q=ai:lingling.zhangSummary: We prove the global-in-time convergence of an Euler-Poisson system near a constant equilibrium state in the whole space \(\mathbb{R}^d\), as physical parameters tend to zero. The result follows from the uniform global existence of smooth solutions by means of energy estimates together with compactness arguments. For this purpose, we establish uniform estimates for \(\operatorname{div} u\) and \(\operatorname{curl} u\) instead of \(\nabla u\).On anisotropic Caginalp phase-field type models with singular nonlinear terms.https://www.zbmath.org/1456.350422021-04-16T16:22:00+00:00"Miranville, Alain"https://www.zbmath.org/authors/?q=ai:miranville.alain-m"Ntsokongo, A. J."https://www.zbmath.org/authors/?q=ai:ntsokongo.armel-judiceSummary: Our aim in this paper is to study the well-posedness and the existence of the global attractor of anisotropic Caginalp phase-field type models with singular nonlinear terms. The main difficulty is to prove, in one and two space dimensions, that the order parameter remains in the physically relevant range and this is achieved by deriving proper a priori estimates.On spectral asymptotics and bifurcation for some elliptic equations of Kirchhoff-type with odd superlinear term.https://www.zbmath.org/1456.351502021-04-16T16:22:00+00:00"Yan, Baoqiang"https://www.zbmath.org/authors/?q=ai:yan.baoqiang"O'Regan, Donal"https://www.zbmath.org/authors/?q=ai:oregan.donal"Agarwal, Ravi P."https://www.zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: In this paper, from estimating the eigenvalues for Kirchhoff elliptic equations, we obtain spectral asymptotics and bifurcation concerning the eigenvalues of some related elliptic linear problem.New exact solutions of a generalised Boussinesq equation with damping term and a system of variant Boussinesq equations via double reduction theory.https://www.zbmath.org/1456.350692021-04-16T16:22:00+00:00"Okeke, Justina Ebele"https://www.zbmath.org/authors/?q=ai:okeke.justina-ebele"Narain, Rivendra"https://www.zbmath.org/authors/?q=ai:narain.rivendra"Govinder, Keshlan Sathasiva"https://www.zbmath.org/authors/?q=ai:govinder.keshlan-sathasivaSummary: The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.Forced oscillation of conformable fractional partial delay differential equations with impulses.https://www.zbmath.org/1456.352202021-04-16T16:22:00+00:00"Saker, S. H."https://www.zbmath.org/authors/?q=ai:saker.samir-h|saker.s-h-ean-tong"Logaarasi, K."https://www.zbmath.org/authors/?q=ai:logaarasi.kandhasamy"Sadhasivam, V."https://www.zbmath.org/authors/?q=ai:sadhasivam.vadivelSummary: In this paper, we establish some interval oscillation criteria for impulsive conformable fractional partial delay differential equations with a forced term. The main results will be obtained by employing Riccati technique. Our results extend and improve some results reported in the literature for the classical differential equations without impulses. An example is provided to illustrate the relevance of the new theorems.Finite time blow-up and global existence of weak solutions for pseudo-parabolic equation with exponential nonlinearity.https://www.zbmath.org/1456.351152021-04-16T16:22:00+00:00"Long, Qunfei"https://www.zbmath.org/authors/?q=ai:long.qunfei"Chen, Jianqing"https://www.zbmath.org/authors/?q=ai:chen.jianqing"Yang, Ganshan"https://www.zbmath.org/authors/?q=ai:yang.ganshanSummary: This paper is concerned with the initial boundary value problem of a class of pseudo-parabolic equation \(u_t - \triangle u - \triangle u_t + u = f(u)\) with an exponential nonlinearity. The eigenfunction method and the Galerkin method are used to prove the blow-up, the local existence and the global existence of weak solutions. Moreover, we also obtain other properties of weak solutions by the eigenfunction method.Global well-posedness of axisymmetric solution to the 3D axisymmetric chemotaxis-Navier-Stokes equations with logistic source.https://www.zbmath.org/1456.351732021-04-16T16:22:00+00:00"Zhang, Qian"https://www.zbmath.org/authors/?q=ai:zhang.qian"Zheng, Xiaoxin"https://www.zbmath.org/authors/?q=ai:zheng.xiaoxinThe system of coupled chemotaxis and incompressible Navier-Stokes equations is studied in the three-dimensional space. The main result is that axisymmetric solutions corresponding to reasonably regular initial data are unique, global-in-time and regular. A novel type apriori inequalities are used in the proofs together with nice symmetry arguments.
Reviewer: Piotr Biler (Wrocław)On the asymptotic dynamics of 2-D magnetic quantum systems.https://www.zbmath.org/1456.811762021-04-16T16:22:00+00:00"Cárdenas, Esteban"https://www.zbmath.org/authors/?q=ai:cardenas.esteban"Hundertmark, Dirk"https://www.zbmath.org/authors/?q=ai:hundertmark.dirk"Stockmeyer, Edgardo"https://www.zbmath.org/authors/?q=ai:stockmeyer.edgardo"Vugalter, Semjon"https://www.zbmath.org/authors/?q=ai:vugalter.semjon-aSummary: In this work, we provide results on the long-time localization in space (dynamical localization) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form \(H = H_0 + W\), where \(H_0\) is rotationally symmetric and has dense point spectrum and \(W\) is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the angular momentum operator in time.Sharp asymptotics for the solutions of the three-dimensional massless Vlasov-Maxwell system with small data.https://www.zbmath.org/1456.351912021-04-16T16:22:00+00:00"Bigorgne, Léo"https://www.zbmath.org/authors/?q=ai:bigorgne.leoSummary: This paper is concerned with the asymptotic properties of the small data solutions to the massless Vlasov-Maxwell system in \(3d\). We use vector field methods to derive almost optimal decay estimates in null directions for the electromagnetic field, the particle density and their derivatives. No compact support assumption in \(x\) or \(v\) is required on the initial data, and the decay in \(v\) is in particular initially optimal. Consistently with Proposition 8.1 of \textit{L. Bigorgne} [``Asymptotic properties of small data solutions of the Vlasov-Maxwell system in high dimensions'', Preprint, \url{arXiv:1712.09698}], the Vlasov field is supposed to vanish initially for small velocities. In order to deal with the slow decay rate of the solutions near the light cone and to prove that the velocity support of the particle density remains bounded away from 0, we make crucial use of the null properties of the system.Random dynamics of non-autonomous fractional stochastic \(p\)-Laplacian equations on \(\mathbb{R}^N\).https://www.zbmath.org/1456.352462021-04-16T16:22:00+00:00"Wang, Renhai"https://www.zbmath.org/authors/?q=ai:wang.renhai"Wang, Bixiang"https://www.zbmath.org/authors/?q=ai:wang.bixiangSummary: This article is concerned with the random dynamics of a wide class of non-autonomous, \textit{non-local}, \textit{ fractional}, stochastic \(p\)-Laplacian equations driven by multiplicative white noise on the entire space \(\mathbb{R}^N\). We first establish the well-posedness of the equations when the time-dependent non-linear drift terms have polynomial growth of arbitrary orders \(p,q\ge 2\). We then prove that the equation has a unique \textit{bi-spatial} pullback random attractor that is \textit{measurable}, compact in \(L^2(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\) and attracts all random subsets of \(L^2(\mathbb{R}^N)\) under the topology of \(L^2(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\). In addition, we establish the upper semi-continuity of these attractors in \(L^2(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\) when the density of noise shrinks to zero. The idea of uniform tail estimates and the method of asymptotic a priori estimates are applied to prove the pullback asymptotic compactness of the solutions in \(L^2(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\) to overcome the non-compactness of Sobolev embeddings on \(\mathbb{R}^N\) as well as the almost sure nondifferentiability of the sample paths of the Wiener process.Period-doubling and Neimark-Sacker bifurcations of a Beddington host-parasitoid model with a host refuge effect.https://www.zbmath.org/1456.352022021-04-16T16:22:00+00:00"Kalabušić, Senada"https://www.zbmath.org/authors/?q=ai:kalabusic.senada"Drino, Džana"https://www.zbmath.org/authors/?q=ai:drino.dzana"Pilav, Esmir"https://www.zbmath.org/authors/?q=ai:pilav.esmirGeneral energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback.https://www.zbmath.org/1456.350352021-04-16T16:22:00+00:00"Li, Fushan"https://www.zbmath.org/authors/?q=ai:li.fushan"Du, Guangwei"https://www.zbmath.org/authors/?q=ai:du.guangweiSummary: In this paper, we consider a degenerate viscoelastic Petrovsky-type plate equation
\[K(\boldsymbol{x})u_{tt}+\Delta^2u-\int_0^tg(t-s)\Delta^2u(s)ds+f(u)=0\]
with boundary feedback. Under the weaker assumption on the relaxation function, the general energy decay is proved by priori estimates and analysis of Lyapunov-like functional. The exponential decay result and polynomial decay result in some literature are special cases of this paper.Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients.https://www.zbmath.org/1456.350182021-04-16T16:22:00+00:00"Gu, Chenlin"https://www.zbmath.org/authors/?q=ai:gu.chenlinSummary: We study the iterative algorithm proposed by Armstrong et al. (An iterative method for elliptic problems with rapidly oscillating coefficients, 2018. arXiv preprint arXiv:1803.03551) to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating coefficients and thus usually require very expensive numerical calculations, while this iterative method is comparatively easy to compute. In this article, we strengthen the estimate for the contraction factor achieved by one iteration of the algorithm. We obtain an estimate that holds uniformly over the initial function in the iteration, and which grows only logarithmically with the size of the domain.Micropolar fluid flows with delay on 2D unbounded domains.https://www.zbmath.org/1456.351682021-04-16T16:22:00+00:00"Sun, Wenlong"https://www.zbmath.org/authors/?q=ai:sun.wenlongSummary: In this paper, we investigate the incompressible micropolar fluid flows on 2D unbounded domains with external force containing some hereditary characteristics. Since Sobolev embeddings are not compact on unbounded domains, first, we investigate the existence and uniqueness of the stationary solution, and further verify its exponential stability under appropriate conditions-essentially the viscosity \(\delta_1:=\min\{\nu, c_a+c_d\}\) is asked to be large enough. Then, we establish the global well-posedness of the weak solutions via the Galerkin method combined with the technique of truncation functions and decomposition of spatial domain.Regularity of homogenized boundary data in periodic homogenization of elliptic systems.https://www.zbmath.org/1456.350232021-04-16T16:22:00+00:00"Shen, Zhongwei"https://www.zbmath.org/authors/?q=ai:shen.zhongwei|shen.zhongwei.1"Zhuge, Jinping"https://www.zbmath.org/authors/?q=ai:zhuge.jinpingThe paper under review deals with the regularity of the homogenized boundary data in periodic homogenization of elliptic systems. More precisely, the authors consider the uniformly elliptic operator with periodically oscillating coefficients
\[
\mathcal{L}_\varepsilon \equiv \mathrm{div}\Big(A(x/\varepsilon)\nabla\Big)=-\frac{\partial}{\partial x_i}\bigg[a_{ij}^{\alpha \beta}(x/\varepsilon)\frac{\partial}{\partial x_j}\bigg]\, ,
\]
where \(1\leq i,j\leq d\), \(1\leq \alpha, \beta \leq m\), \(0<\varepsilon \leq 1\) and the coefficient matrix \(A\) satisfies ellipticity, periodicity, and smoothness assumptions. Then, for example, they introduce the Dirichlet problem with oscillating boundary data
\[
\left\{
\begin{array}{ll}
\mathcal{L}_\varepsilon(u_\varepsilon)=0 & \mathrm{in}\ \Omega\, ,\\
u_\varepsilon(x)=f(x,x/\varepsilon) & \mathrm{on}\ \partial \Omega\, ,
\end{array}
\right.
\]
where \(f\) is \(1\)-periodic in \(y\). Under suitable assumptions, the solution is well-known to converge to the solution \(u_0\) of the homogenized problem
\[
\left\{
\begin{array}{ll}
\mathcal{L}_0(u_0)=0 & \mathrm{in}\ \Omega\, ,\\
u_0=\overline{f} & \mathrm{on}\ \partial \Omega\, ,
\end{array}
\right.
\]
where \(\mathcal{L}_0\) is the usual homogenized operator and \(\overline{f}\) is a function whose value at \(x \in \partial \Omega\) depends only on \(A\), \(f(x,\cdot)\) and the outward normal \(n\) to \(\partial \Omega\) at \(x\). The authors prove regularity results for the homogenized datum \(\overline{f}\), namely that it belongs to \(W^{1,p}\) for any \(1<p<+\infty\). Similar results are obtained also for the homogenized datum of the Neumann problem.
Reviewer: Paolo Musolino (Padova)The Liouville theorem for \(p\)-harmonic functions and quasiminimizers with finite energy.https://www.zbmath.org/1456.350552021-04-16T16:22:00+00:00"Björn, Anders"https://www.zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://www.zbmath.org/authors/?q=ai:bjorn.jana"Shanmugalingam, Nageswari"https://www.zbmath.org/authors/?q=ai:shanmugalingam.nageswariSummary: We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite \(p\)th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global \(p\)-Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line \(\mathbf{R}\), we characterize all locally doubling measures, supporting a local \(p\)-Poincaré inequality, for which there exist nonconstant quasiminimizers of finite \(p\)-energy, and show that a quasiminimizer is of finite \(p\)-energy if and only if it is bounded. As \(p\)-harmonic functions are quasiminimizers they are covered by these results.Global regularity for 2D fractional magneto-micropolar equations.https://www.zbmath.org/1456.351672021-04-16T16:22:00+00:00"Shang, Haifeng"https://www.zbmath.org/authors/?q=ai:shang.haifeng"Wu, Jiahong"https://www.zbmath.org/authors/?q=ai:wu.jiahongSummary: The magneto-micropolar equations are important models in fluid mechanics and material sciences. This paper focuses on the global regularity problem on the 2D magneto-micropolar equations with fractional dissipation. We establish the global regularity for three important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ various technics including the regularization of generalized heat operators on the Fourier frequency localized functions, logarithmic Sobolev interpolation inequalities and the maximal regularity property of the heat operator.Singularity formation for the fractional Euler-alignment system in 1D.https://www.zbmath.org/1456.351592021-04-16T16:22:00+00:00"Arnaiz, Victor"https://www.zbmath.org/authors/?q=ai:arnaiz.victor"Castro, Ángel"https://www.zbmath.org/authors/?q=ai:castro.angelThe Euler alignment system is a microscopic version of the Cucker-Smale model of the collective behavior of agents. In the case of a power-like influence function which describes the velocity alignment strength between agents, this can be reduced to the fractional partial differential equation
\[u_t+(u\Lambda^{\alpha-1}Hu)_x=0\]
(occurring also in the dislocation density evolution in solids), where \(\Lambda^{\alpha-1}v(x)=c_\alpha\int_{\mathbb R}\frac{v(y)}{|x-y|^\alpha}dy\), and \(H\) is the Hilbert transform.
The authors prove that the formation of singularities in a finite time occur for that equation for solutions of the Cauchy problem and some smooth initial data in case \(1<\alpha<2\), as well as in case \(0<\alpha<1\) when the space-periodic problem is studied. Here, the singularities are understood in \(H^{3/2+\alpha}\) and \(C^1\) sense, respectively.
Reviewer: Piotr Biler (Wrocław)Global well-posedness for the 2D fractional Boussinesq equations in the subcritical case.https://www.zbmath.org/1456.351742021-04-16T16:22:00+00:00"Zhou, Daoguo"https://www.zbmath.org/authors/?q=ai:zhou.daoguo"Li, Zilai"https://www.zbmath.org/authors/?q=ai:li.zilai"Shang, Haifeng"https://www.zbmath.org/authors/?q=ai:shang.haifeng"Wu, Jiahong"https://www.zbmath.org/authors/?q=ai:wu.jiahong"Yuan, Baoquan"https://www.zbmath.org/authors/?q=ai:yuan.baoquan"Zhao, Jiefeng"https://www.zbmath.org/authors/?q=ai:zhao.jiefengThe paper deals with the global in time well-posedness problem for the 2D Boussinesq equations with fractional dissipation in the subcritical regime. Namely, in the velocity dissipation dominated case for the largest possible range of the exponent of \(-\Delta^{\frac{\alpha}{2}}\), \(\alpha\in(0,1)\), the existence of a unique regular global in time solution is proved.
Reviewer: Georg V. Jaiani (Tbilisi)An order approach to SPDEs with antimonotone terms.https://www.zbmath.org/1456.352452021-04-16T16:22:00+00:00"Scarpa, Luca"https://www.zbmath.org/authors/?q=ai:scarpa.luca"Stefanelli, Ulisse"https://www.zbmath.org/authors/?q=ai:stefanelli.ulisseSummary: We consider a class of parabolic stochastic partial differential equations featuring an antimonotone nonlinearity. The existence of unique maximal and minimal variational solutions is proved via a fixed-point argument for nondecreasing mappings in ordered spaces. This relies on the validity of a comparison principle.Dynamical transition theory of hexagonal pattern formations.https://www.zbmath.org/1456.371032021-04-16T16:22:00+00:00"Şengül, Taylan"https://www.zbmath.org/authors/?q=ai:sengul.taylanSummary: The main goal of this paper is to understand the formation of hexagonal patterns from the dynamical transition theory point of view. We consider the transitions from a steady state of an abstract nonlinear dissipative system. To shed light on the formation of mixed mode patterns such as the hexagonal pattern, we consider the case where the linearized operator of the system has two critical real eigenvalues, at a critical value \(\lambda_c\) of a control parameter \(\lambda\) with associated eigenmodes having a roll and rectangular pattern. By using center manifold reduction, we obtain the reduced equations of the system near the critical transition value \(\lambda_c\). By a through analysis of these equations, we fully characterize all possible transition scenarios when the coefficients of the quadratic part of the reduced equations do not vanish. We consider three problems, two variants of the 2D Swift-Hohenberg equation and the 3D surface tension driven convection, to demonstrate that all the main theoretical results we obtain here are indeed realizable.Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model.https://www.zbmath.org/1456.350622021-04-16T16:22:00+00:00"Xu, Xiangsheng"https://www.zbmath.org/authors/?q=ai:xu.xiangshengSummary: In this paper we study partial regularity of weak solutions to the initial boundary value problem for the system \(-\operatorname{div}\left[ (I+\mathbf{m}\otimes \mathbf{m})\nabla p\right] =S(x), \partial_t\mathbf{m}-D^2\Delta \mathbf{m}-E^2(\mathbf{m}\cdot \nabla p)\nabla p+|\mathbf{m}|^{2(\gamma -1)}{} \mathbf{m}=0\), where \(S(x)\) is a given function and \(D, E, \gamma\) are given numbers. This problem has been proposed as a PDE model for biological transportation networks. The mathematical difficulty is due to the fact that the system in the model features both a quadratic nonlinearity and a cubic nonlinearity. The regularity issue seems to have a connection to a conjecture by De Giorgi (Congetture sulla continuitá delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati, Unpublished, 1995). We also investigate the life-span of classical solutions. Our results show that local existence of a classical solution can always be obtained and the life-span of such a solution can be extended as far away as one wishes as long as the term \(\Vert \mathbf{m}(x,0)\Vert_{\infty, \Omega}+\Vert S(x)\Vert_{\frac{2N}{3}, \Omega}\) is made suitably small, where \(N\) is the space dimension and \(\Vert \cdot \Vert_{q,\Omega}\) denotes the norm in \(L^q(\Omega)\).Polynomial decay rate for a coupled Lamé system with viscoelastic damping and distributed delay terms.https://www.zbmath.org/1456.351322021-04-16T16:22:00+00:00"Doudi, Nadjat"https://www.zbmath.org/authors/?q=ai:doudi.nadjat"Boulaaras, Salah Mahmoud"https://www.zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Alghamdi, Ahmad Mohammed"https://www.zbmath.org/authors/?q=ai:alghamdi.ahmad-m-a"Cherif, Bahri"https://www.zbmath.org/authors/?q=ai:cherif.bahriSummary: In this paper, we prove a general energy decay results of a coupled Lamé system with distributed time delay. By assuming a more general of relaxation functions and using some properties of convex functions, we establish the general energy decay results to the system by using an appropriate Lyapunov functional.Type II blowup in a doubly parabolic Keller-Segel system in two dimensions.https://www.zbmath.org/1456.350482021-04-16T16:22:00+00:00"Mizoguchi, Noriko"https://www.zbmath.org/authors/?q=ai:mizoguchi.norikoSummary: This paper is concerned with a parabolic-parabolic Keller-Segel system\[\begin{cases} u_t = \operatorname{\nabla} \cdot(\operatorname{\nabla} u - u \operatorname{\nabla} v) \qquad & \text{ in } \operatorname{\Omega} \times(0, T), \\ v_t = \operatorname{\Delta} v - v + u & \text{ in } \operatorname{\Omega} \times(0, T), \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 & \text{ on } \partial \operatorname{\Omega} \times(0, T), \\ u(x, 0) = u_0(x) \geq 0, v(x, 0) = v_0(x) \geq 0 & \text{ in } \operatorname{\Omega} \end{cases}\] in a smoothly bounded domain \(\operatorname{\Omega} \subset \mathbb{R}^2\). Keller and Segel proposed it as a model of aggregation of bacteria, which is mathematically translated as finite-time blowup. A solution \((u, v)\) is said to blow up at \(t = T < + \infty\) if \(\lim \sup_{t \rightarrow T} \| u(t) \|_{L^\infty(\operatorname{\Omega})} = + \infty\). When \((u, v)\) blows up at \(t = T\), the blowup is called type I if \(\| u(t) \|_{L^\infty(\operatorname{\Omega})} \leq C(T - t)^{- 1}\) for \(t \in [0, T)\) with some constant \(C > 0\), and type II otherwise. In spite of motivation of the modeling, it has remained open for the past decades whether or not finite-time blowup occurs frequently in the parabolic-parabolic system. Recently it was shown in that there exists a large class of radial initial data such that the corresponding solutions blow up in finite time. However they did not give any information on blowup behavior. Blowup rate is one of the greatest concerns in investigation of blowup mechanism. In this paper, we show that each blowup is of type II in the parabolic-parabolic system in radial case.On the stochastic Dullin-Gottwald-Holm equation: global existence and wave-breaking phenomena.https://www.zbmath.org/1456.601612021-04-16T16:22:00+00:00"Rohde, Christian"https://www.zbmath.org/authors/?q=ai:rohde.christian"Tang, Hao"https://www.zbmath.org/authors/?q=ai:tang.haoSummary: We consider a class of stochastic evolution equations that include in particular the stochastic Camassa-Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces \(H^s\) with \(s>3/2\). Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.Global existence and decay in multi-component reaction-diffusion-advection systems with different velocities: oscillations in time and frequency.https://www.zbmath.org/1456.351092021-04-16T16:22:00+00:00"de Rijk, Björn"https://www.zbmath.org/authors/?q=ai:de-rijk.bjorn"Schneider, Guido"https://www.zbmath.org/authors/?q=ai:schneider.guido.1|schneider.guidoSummary: It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of solutions with small, localized initial data and even finite time blow-up. It was recently proved, however, that, if the components of two nonlinearly coupled reaction-diffusion-advection equations propagate with different velocities, then quadratic or cubic mixed-terms, i.e. nonlinear terms with nontrivial contributions from both components, do not affect global existence and Gaussian decay of small, localized initial data. The proof relied on pointwise estimates to capture the difference in velocities. In this paper we present an alternative method, which is better applicable to multiple components. Our method involves a nonlinear iteration scheme that employs \(L^1-L^p\) estimates in Fourier space and exploits oscillations in time and frequency, which arise due to differences in transport. Under the assumption that each component exhibits different velocities, we establish global existence and decay for small, algebraically localized initial data in multi-component reaction-diffusion-advection systems allowing for cubic mixed-terms and nonlinear terms of Burgers' type.Harnack inequalities and heat kernel estimates for degenerate diffusion operators arising in population biology.https://www.zbmath.org/1456.921132021-04-16T16:22:00+00:00"Epstein, Charles L."https://www.zbmath.org/authors/?q=ai:epstein.charles-l"Mazzeo, Rafe"https://www.zbmath.org/authors/?q=ai:mazzeo.rafe-rSummary: This paper continues the analysis, started in our works [SIAM J. Math. Anal. 42, No. 2, 568--608 (2010; Zbl 1221.35063); Degenerate diffusion operators arising in population biology. Princeton, NJ: Princeton University Press (2013; Zbl 1309.47001)], of a class of degenerate elliptic operators defined on manifolds with corners, which arise in Population Biology. Using techniques pioneered by J. Moser, and extended and refined by L. Saloff-Coste, A. Grigor'yan, and K.-T. Sturm, we show that weak solutions to the parabolic problem defined by a subclass of these operators, which consists of those that can be defined by Dirichlet forms and have non-vanishing transverse vector field, satisfy a Harnack inequality. This allows us to conclude that the solutions to these equations belong, for positive times, to the natural anisotropic Hölder spaces, and also leads to upper and, in some cases, lower bounds for the heat kernels of these operators. These results imply that these operators have a compact
resolvent when acting on \(\mathcal{C}^0\) or \(L^2\). The proof relies upon a scale-invariant Poincaré inequality that we establish for a large class of weighted Dirichlet forms, as well as estimates to handle certain mildly singular perturbation terms. The weights that we consider are neither Ahlfors regular nor do they generally belong to the Muckenhaupt class \(A_2\).Higher-order pathwise theory of fluctuations in stochastic homogenization.https://www.zbmath.org/1456.350172021-04-16T16:22:00+00:00"Duerinckx, Mitia"https://www.zbmath.org/authors/?q=ai:duerinckx.mitia"Otto, Felix"https://www.zbmath.org/authors/?q=ai:otto.felixSummary: We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order \(\frac{d}{2} \), where \(d\) is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the ``homogenization commutator'', thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderón-Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution.https://www.zbmath.org/1456.350262021-04-16T16:22:00+00:00"Ikeda, Masahiro"https://www.zbmath.org/authors/?q=ai:ikeda.masahiro"Tanaka, Tomoyuki"https://www.zbmath.org/authors/?q=ai:tanaka.tomoyuki"Wakasa, Kyouhei"https://www.zbmath.org/authors/?q=ai:wakasa.kyouheiIn this paper, the authors study the long time behavior for solutions to the Cauchy problem for the wave equation with a time-dependent critical damping and a cubic convolution \(\partial_t^2 v-\Delta v+\frac{\mu}{1+t}\partial_t v=(V*|v|^2)v\), with \(\mu=2 \), \(V=|x|^{-\gamma}\), \(\gamma\in \left(-\frac{1}{2},3\right)\) and \(x\in \mathbb{R}^3\). For small, nontrivial \(C^3_c\times C_c^2\) data of size \(\varepsilon\), it is shown that the lifespan (denoted by \(T_\varepsilon\)) for the problem satisfies \(T_\varepsilon=\infty\) for \(\gamma\in(0,3)\), \(T_\varepsilon\ge \exp(c \varepsilon^{-2})\) for \(\gamma=0\), and \(c \varepsilon^{2/\gamma}\le T_\varepsilon\le C\varepsilon^{-\delta+2/\gamma}\) for \(\gamma\in (-1/2,0)\) and \(\delta>0\). In particular, it shows the critical power for the problem to admit small data global existence is \(\gamma_c=0\), when \(\mu=2\). The reason for the specific damping constant \(\mu= 2\) lies on the fact that, in this case, a simple change of variable (the Liouville transform) \(v\to u=(1+t)v\) will transform the problem into wave equations without damping \(\partial_t^2 u-\Delta u=(1+t)^{-2}(V*|u|^2)u\).
Reviewer: Chengbo Wang (Hangzhou)Existence of a ground state and blowup problem for a class of nonlinear Schrödinger equations involving mass and energy critical exponents.https://www.zbmath.org/1456.350472021-04-16T16:22:00+00:00"Kikuchi, Hiroaki"https://www.zbmath.org/authors/?q=ai:kikuchi.hiroaki"Watanabe, Minami"https://www.zbmath.org/authors/?q=ai:watanabe.minamiSummary: In this paper, we study the existence of the ground state and blowup problem for a class of nonlinear Schrödinger equations involving the mass and energy critical exponents. To show the existence of a ground state, we solve a minimization problem related to the virial identity, so that we need to compare the minimization value to the best constant of the Gagliardo-Nirenberg inequality because our nonlinearities contain the mass critical nonlinearity. Once we obtain the ground state, we can introduce a subset \(\mathcal{A}_{\omega,-}\) of \(H^1(\mathbb{R}^d)\) for each \(\omega >0\) as in [\textit{H. Berestycki} and \textit{T. Cazenave}, C. R. Acad. Sci., Paris, Sér. I 293, 489--492 (1981; Zbl 0492.35010)]. Then, it turn out that any radial solution starting from \(\mathcal{A}_{\omega,-}\) blows up in a finite time.Second order estimates for Hessian equations of parabolic type on Riemannian manifolds.https://www.zbmath.org/1456.351202021-04-16T16:22:00+00:00"Jiao, Heming"https://www.zbmath.org/authors/?q=ai:jiao.hemingSummary: In this paper, we establish the second order estimates for solutions of the first initial-boundary value problem for general Hessian type fully nonlinear parabolic equations on Riemannian manifolds. The techniques used in this article can work for a wide range of fully nonlinear PDEs under very general conditions.Book review of: A. Miranville, The Cahn-Hilliard equation: recent advances and applications.https://www.zbmath.org/1456.000042021-04-16T16:22:00+00:00"Abels, Helmut"https://www.zbmath.org/authors/?q=ai:abels.helmutReview of [Zbl 1446.35001].Global regular solutions to a Kelvin-Voigt type thermoviscoelastic system.https://www.zbmath.org/1456.740322021-04-16T16:22:00+00:00"Pawłow, Irena"https://www.zbmath.org/authors/?q=ai:pawlow.irena"Zajączkowski, Wojciech M."https://www.zbmath.org/authors/?q=ai:zajaczkowski.wojciech-mGlobal large solutions to the three dimensional compressible Navier-Stokes equations.https://www.zbmath.org/1456.351712021-04-16T16:22:00+00:00"Zhai, Xiaoping"https://www.zbmath.org/authors/?q=ai:zhai.xiaoping"Li, Yongsheng"https://www.zbmath.org/authors/?q=ai:li.yongsheng"Zhou, Fujun"https://www.zbmath.org/authors/?q=ai:zhou.fujunThe authors study the global existence of some class of large strong solutions to the three-dimensional compressible
Navier-Stokes equations with density-dependent viscosity coefficients in the critical Besov spaces. Applying
the weighted Chemin-Lerner technique used for the incompressible Navier-Stokes equations
[\textit{J.-Y. Chemin} and \textit{I. Gallagher}, Trans. Am. Math. Soc. 362, No. 6, 2859--2873 (2010; Zbl 1189.35220)] and using the smooth effect for the heat
flow, the authors carefully employ energy estimates to establish the uniform a priori estimates, with the help of which
a local strong solution can be continued globally in time, provided that the initial data satisfy some nonlinear smallness condition,
but any component of the initial velocity could be arbitrarily large.
Moreover, the authors give an example of initial data satisfying the nonlinear smallness condition,
while the norm of each component of the initial velocity can be arbitrarily large.
Reviewer: Song Jiang (Beijing)Touchdown is the only finite time singularity in a three-dimensional MEMS model.https://www.zbmath.org/1456.351032021-04-16T16:22:00+00:00"Laurençot, Philippe"https://www.zbmath.org/authors/?q=ai:laurencot.philippe"Walker, Christoph"https://www.zbmath.org/authors/?q=ai:walker.christophSummary: Touchdown is shown to be the only possible finite time singularity that may take place in a free boundary problem modeling a three-dimensional microelectromechanical system. The proof relies on the energy structure of the problem and uses smoothing effects of the semigroup generated in \(L_1\) by the bi-Laplacian with clamped boundary conditions.Propagation of smallness in elliptic periodic homogenization.https://www.zbmath.org/1456.350212021-04-16T16:22:00+00:00"Kenig, Carlos"https://www.zbmath.org/authors/?q=ai:kenig.carlos-e"Zhu, Jiuyi"https://www.zbmath.org/authors/?q=ai:zhu.jiuyiThe paper under review deals with the propagation of smallness in elliptic periodic homogenization, in other words with the issue of how a solution \(u\) of a second order PDE \(Lu=0\) on a domain \(X\) can be made arbitrary small on any given compact subset of \(X\) by making it sufficiently small on an arbitrary given subdomain \(Y\). An example of quantitative propagation of smallness is given by the so-called three-ball inequality. The authors work in the setting of elliptic periodic homogenization and to do so they introduce a family of elliptic operators in divergence form with rapidly oscillating periodic coefficients
\[
\mathcal{L}_\epsilon u_\epsilon =-\mathrm{div}\Big(A(x/\epsilon)\nabla u_\epsilon\Big)=0\text{ in }\Omega\, ,
\]
where \(\epsilon>0\), \(\Omega\) is a bounded subdomain in \(\mathbb{R}^d\) (\(d \geq 2\)), and the matrix \(A\) satisfies ellipticity, periodicity, and Hölder continuity assumptions. Then for the solutions to \(\mathcal{L}_\epsilon u_\epsilon =0\) the authors obtain approximate three-ball inequalities in ellipsoids.
Reviewer: Paolo Musolino (Padova)Global existence of weak solutions for the anisotropic compressible Stokes system.https://www.zbmath.org/1456.351602021-04-16T16:22:00+00:00"Bresch, Didier"https://www.zbmath.org/authors/?q=ai:bresch.didier"Burtea, C."https://www.zbmath.org/authors/?q=ai:burtea.cosminThe authors consider the quasi-stationary compressible Stokes equations \(\partial _{t}\rho +div(\rho u)=0\), \(-\operatorname{div}\tau +a\nabla \rho^{\gamma }=f\), \(\gamma >1\), where \(u\) is the fluid velocity field, \(\rho \) the fluid density and \(\tau \) the viscous stress tensor defined through \(\tau _{ij}(t,x,D(u))=A_{ijkl}(t,x)D[u]_{kl}\), for a matrix \(A_{ijkl}(t,x)\in W^{1,\infty }((0,T)\times \mathbb{T}^{3})\). The initial condition \(\rho \mid_{t=0}=\rho _{0}\geq 0\) is imposed. The main result of the paper proves that if \(f\in W^{1,2}(0,T;L^{6/5}(\mathbb{T}^{3})\) satisfies \(\int_{\mathbb{T}^{3}}f(t)dx=0\), if \(\rho _{0}\geq 0\) further satisfies \(0<\int_{\mathbb{T}^{3}}\rho _{0}dx<+\infty \) and \(\int_{\mathbb{T}^{3}}\rho _{0}^{\gamma}dx<+\infty \), there exists a global weak solution \((\rho ,u)\) to the above system with \(\rho \in C([0,T];L_{weak}^{\gamma }(\mathbb{T}^{3}))\cap L^{2\gamma }((0,T)\times \mathbb{T}^{3})\), \(u\in L^{2}((0,T);H^{1}(\mathbb{T}
^{3}))\) and \(\int_{\mathbb{T}^{3}}udx=0\). The authors assume that \(A\) satisfies some symmetry properties and that the tensor \(\tau \) satisfies regularity hypotheses. For the proof, the authors first define a finite-energy weak solution to the problem \(\partial _{t}\rho +\operatorname{div}(\rho u)=0\), \(-\operatorname{div}\tau +\nabla \rho ^{\gamma }=f\), \(\rho \mid_{t=0}=\rho _{0}\geq 0\), as a pair \((\rho ,u)\in L^{\infty }(0,T;L^{\gamma }(\mathbb{T}^{3}))\times L^{2}(0,T;H^{1}(\mathbb{T}^{3}))\) which satisfies this problem in the sense of distributions, the mass conservation identity \(\int_{\mathbb{T}^{3}}\rho (t)=\int_{\mathbb{T}^{3}}\rho _{0}\) and the energy inequality \(\int_{\mathbb{T}^{3}}\rho ^{\gamma }+\int_{0}^{t}\int_{\mathbb{T}^{3}}\tau :\nabla udx\leq \int_{\mathbb{T}^{3}}\rho _{0}^{\gamma
}+\int_{0}^{t}\int_{\mathbb{T}^{3}}u\cdot f\). They prove estimates and a stability property for sequences of finite-energy weak solutions, the limit satisfying an identity which involves a defect measure associated to the pressure. For the construction of solutions to the original problem, the authors introduce a regularized system with diffusion and drag terms in the density equation: \(\partial _{t}\rho +\operatorname{div}(\rho \omega _{\delta }\ast u)=\varepsilon \Delta \rho -\varepsilon \rho ^{2\gamma }-\varepsilon \rho^{2\gamma +1}-\varepsilon \rho ^{3}\), \(\mathcal{A}u+\nabla \omega _{\delta}\ast \rho ^{\gamma }=f\), where \(\omega _{\delta }\) is a regularizing kernel
and \(\mathcal{A}\) is the application \(v\rightarrow -div\tau (t,x,D(v))\). The authors prove the global existence and uniqueness of a strong solution on \((0,T)\) using a fixed point argument. They pass to the limit with respect to the regularization parameter \(\delta \), then with respect to \(\varepsilon \), thanks to estimates and a weak stability result for the perturbed system, to obtain a global solution to the quasi-stationary compressible Stokes system. The paper ends with possible extensions of the main result to close systems.
Reviewer: Alain Brillard (Riedisheim)Competition instabilities of spike patterns for the 1D Gierer-Meinhardt and Schnakenberg models are subcritical.https://www.zbmath.org/1456.350132021-04-16T16:22:00+00:00"Kolokolnikov, Theodore"https://www.zbmath.org/authors/?q=ai:kolokolnikov.theodore"Paquin-Lefebvre, Frédéric"https://www.zbmath.org/authors/?q=ai:paquin-lefebvre.frederic"Ward, Michael J."https://www.zbmath.org/authors/?q=ai:ward.michael-jContinuous and discrete periodic asymptotic behavior of solutions to a competitive chemotaxis PDEs system.https://www.zbmath.org/1456.350362021-04-16T16:22:00+00:00"Negreanu, M."https://www.zbmath.org/authors/?q=ai:negreanu.mihaela"Vargas, A. M."https://www.zbmath.org/authors/?q=ai:vargas.antonio-manuelSummary: In this paper we study the continuous and full discrete versions of a parabolic-parabolic-elliptic system with periodic terms that serves as a model for some chemotaxis phenomena. This model appears naturally in the interaction of two biological species and a chemical. The presence of the periodic terms has a strong impact on the behavior of the solutions. Some conditions on the system's data are given that guarantee the global existence of solutions that converge to periodical solutions of an associated ODE's system. Further, we analyze the discretized version of the model using a Generalized Finite Difference Method (GFDM) and we confirm that the properties of the continuous model are also preserved for the resulting discrete model. To this end, we prove the conditional convergence of the numerical model and study some practical examples.Global Hölder estimates via Morrey norms for hypoelliptic operators with drift.https://www.zbmath.org/1456.350522021-04-16T16:22:00+00:00"Hou, Yuexia"https://www.zbmath.org/authors/?q=ai:hou.yuexia"Niu, Pengcheng"https://www.zbmath.org/authors/?q=ai:niu.pengchengSummary: Suppose that \(X_0, X_1, \ldots, X_m\) are left invariant real vector fields on the homogeneous group \(G\) with \(X_0\) being homogeneous of degree two and \(X_1, \ldots, X_m\) homogeneous of degree one. In the paper we study the hypoelliptic operator with drift of the kind \(L = \sum_{i, j = 1}^m a_{i j} X_i X_j + a_0 X_0,\) where \(a_0 \neq 0\) and \((a_{i j})\) is a constant matrix satisfying the elliptic condition on \(\mathbb{R}^m\). By proving the boundedness of two integral operators on the Morrey spaces with two weights, we obtain global Hölder estimates for \(L\).Localization of the vorticity direction conditions for the 3D shear thickening fluids.https://www.zbmath.org/1456.351562021-04-16T16:22:00+00:00"Yang, Jiaqi"https://www.zbmath.org/authors/?q=ai:yang.jiaqiSummary: It is obtained that a localization of the vorticity direction coherence conditions for the regularity of the 3D shear thickening fluids to an arbitrarily small space-time cylinder. It implies the regularity of any geometrically constrained weak solution of the system considered independently of the type of the spatial domain or the boundary conditions.A vanishing dynamic capillarity limit equation with discontinuous flux.https://www.zbmath.org/1456.350122021-04-16T16:22:00+00:00"Graf, M."https://www.zbmath.org/authors/?q=ai:graf.melanie"Kunzinger, M."https://www.zbmath.org/authors/?q=ai:kunzinger.michael"Mitrovic, D."https://www.zbmath.org/authors/?q=ai:mitrovic.dragisa|mitrovic.darko|mitrovich.dushan|mitrovic.dejan|mitrovic.danijela|mitrovic.dorde"Vujadinovic, D."https://www.zbmath.org/authors/?q=ai:vujadinovic.djordjijeSummary: We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the \textbf{dynamics capillarity} equation
\[
\begin{aligned}{\left\{\begin{array}{ll}\partial_t u_{\varepsilon ,\delta }+\text{div} {\mathfrak f}_{\varepsilon ,\delta }(\mathbf{x}, u_{\varepsilon ,\delta })=\varepsilon \Delta u_{\varepsilon ,\delta }+\delta (\varepsilon ) \partial_t \Delta u_{\varepsilon ,\delta }, \mathbf{x} \in M, t\ge 0\\ u|_{t=0}=u_0(\mathbf{x}). \end{array}\right. } \end{aligned}
\]
Here, \(\mathfrak{f}_{\varepsilon ,\delta }\) and \(u_0\) are smooth functions while \(\varepsilon\) and \(\delta =\delta (\varepsilon )\) are fixed constants. Assuming \(\mathfrak{f}_{\varepsilon ,\delta } \rightarrow\mathfrak{f}\in L^p(\mathbb{R}^d\times\mathbb{R};\mathbb{R}^d)\) for some \(1< p< \infty\), strongly as \(\varepsilon \rightarrow 0\), we prove that, under an appropriate relationship between \(\varepsilon\) and \(\delta (\varepsilon )\) depending on the regularity of the flux \(\mathfrak{f}\), the sequence of solutions \((u_{\varepsilon ,\delta })\) strongly converges in \(L^1_{loc}(\mathbb{R}^+\times\mathbb{R}^d)\) toward a solution to the conservation law
\[
\begin{aligned} \partial_t u +\text{div} {{\mathfrak{f}}}(\mathbf{x}, u)=0. \end{aligned}
\]
The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.Positive Liouville theorem and asymptotic behaviour for \((p,A)\)-Laplacian type elliptic equations with Fuchsian potentials in Morrey space.https://www.zbmath.org/1456.350562021-04-16T16:22:00+00:00"Giri, Ratan Kr."https://www.zbmath.org/authors/?q=ai:giri.ratan-kr"Pinchover, Yehuda"https://www.zbmath.org/authors/?q=ai:pinchover.yehudaSummary: We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point \(\zeta \in \partial \Omega \cup \{\infty\}\) of the quasilinear elliptic equation
\[-\mathrm{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad \text{in } \Omega \setminus \{\zeta\},
\]
where \(\Omega\) is a domain in \(\mathbb{R}^d (d\geq 2)\), and \(A=(a_{ij})\in L_{\mathrm{loc}}^\infty (\Omega ;\mathbb{R}^{d\times d})\) is a symmetric and locally uniformly positive definite matrix. The potential \(V\) lies in a certain local Morrey space (depending on \(p)\) and has a Fuchsian-type isolated singularity at \(\zeta\).A fairly strong stability result for parabolic quasiminimizers.https://www.zbmath.org/1456.350252021-04-16T16:22:00+00:00"Fujishima, Yohei"https://www.zbmath.org/authors/?q=ai:fujishima.yohei"Habermann, Jens"https://www.zbmath.org/authors/?q=ai:habermann.jens"Masson, Mathias"https://www.zbmath.org/authors/?q=ai:masson.mathiasSummary: In this paper we consider parabolic \(Q\)-quasiminimizers related to the \(p\)-Laplace equation in \(\Omega_T := \Omega \times (0,T)\). In particular, we focus on the stability problem with respect to the parameters \(p\) and \(Q\). It is known that, if \(Q\rightarrow 1\), then parabolic quasiminimizers with fixed initial-boundary data on \(Q_T\) converge to the parabolic minimizer strongly in \(L^p(0,T;W^{1,p}(\Omega))\) under suitable further structural assumptions. Our concern is whether or not we can obtain even stronger convergence. We will show a fairly strong stability result.On the Cauchy problem for the generalized weak disspative Camassa-Holm equation.https://www.zbmath.org/1456.351792021-04-16T16:22:00+00:00"Chen, Wenxia"https://www.zbmath.org/authors/?q=ai:chen.wenxia"Zhang, Jianmei"https://www.zbmath.org/authors/?q=ai:zhang.jianmei"Deng, Xiaoyan"https://www.zbmath.org/authors/?q=ai:deng.xiaoyan(no abstract)Interactions among different types of nonlinear waves described by the Kadomtsev-Petviashvili equation.https://www.zbmath.org/1456.351772021-04-16T16:22:00+00:00"Cheng, Xue-Ping"https://www.zbmath.org/authors/?q=ai:cheng.xueping"Chen, Chun-Li"https://www.zbmath.org/authors/?q=ai:chen.chunli"Lou, S. Y."https://www.zbmath.org/authors/?q=ai:lou.senyueSummary: In nonlinear science, the interactions among solitons are well studied because the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlevé waves. In this paper, taking the Kadomtsev-Petviashvili (KP) equation as an illustration model, a new method is established to find interactions among different types of nonlinear waves. The nonlocal symmetries related to the Darboux transformation (DT) of the KP equation is localized after embedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions. It is shown that the essential and unique role of the DT is to add an additional soliton on a Boussinesq-type wave or a KdV-type wave, which are two basic reductions of the KP equation.Qualitative properties of weak solutions for \(p\)-Laplacian equations with nonlocal source and gradient absorption.https://www.zbmath.org/1456.351182021-04-16T16:22:00+00:00"Chaouai, Zakariya"https://www.zbmath.org/authors/?q=ai:chaouai.zakariya"El Hachimi, Abderrahmane"https://www.zbmath.org/authors/?q=ai:el-hachimi.abderrahmaneSummary: We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source \[\dfrac{\partial u}{\partial t} -\operatorname{div}(|\nabla u|^{p-2}\nabla u) =\lambda u^k\int_{\Omega}u^sdx- \mu u^l|\nabla u|^q\] in a bounded domain \(\Omega\subset\mathbb{R}^N \), where \(p>1\), the parameters \(k,s,l,q,\lambda>0\) and \(\mu\geq 0\). Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on \(|\nabla u|\). Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.Precursors for waves in random media.https://www.zbmath.org/1456.740092021-04-16T16:22:00+00:00"Bal, Guillaume"https://www.zbmath.org/authors/?q=ai:bal.guillaume"Pinaud, Olivier"https://www.zbmath.org/authors/?q=ai:pinaud.olivier"Ryzhik, Lenya"https://www.zbmath.org/authors/?q=ai:ryzhik.lenya"Sølna, Knut"https://www.zbmath.org/authors/?q=ai:solna.knutSummary: We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O'Doherty-Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully three-dimensional situation.On the Lotka-Volterra competition system with dynamical resources and density-dependent diffusion.https://www.zbmath.org/1456.351072021-04-16T16:22:00+00:00"Wang, Zhi-An"https://www.zbmath.org/authors/?q=ai:wang.zhian"Xu, Jiao"https://www.zbmath.org/authors/?q=ai:xu.jiaoSummary: In this paper, we consider the following Lotka-Volterra competition system with dynamical resources and density-dependent diffusion
\[
\begin{aligned}
\begin{cases}
u_t=\Delta(d_1(w)u)+u(a_1w-b_1u-c_1v),\quad & x\in\Omega,\quad t> 0,\\
v_t=\Delta(d_2(w)v)+v(a_2w-b_2u-c_2v),& x\in\Omega, \quad t>0,\\
w_t=\Delta w-w(u+v)+\mu w(m(x)-w),& x\in\Omega, \quad t> 0,
\end{cases}
\end{aligned} \tag{\(\ast\)}
\]
in a bounded smooth domain \(\Omega\subset\mathbb{R}^2\) with homogeneous Neumann boundary conditions, where the parameters \(\mu,a_i,b_i,c_i (i=1,2)\) are positive constants, \(m(x)\) is the prey's resource, and the dispersal rate function \(d_i(w)\) satisfies the the following hypothesis:
\begin{itemize}
\item \(d_i(w)\in C^2([0,\infty)),d_i'(w)\le 0\) on \([0,\infty)\) and \(d(w)>0\).
\end{itemize}
When \(m(x)\) is constant, we show that the system (*) with has a unique global classical solution when the initial datum is in functional space \(W^{1,p}(\Omega)\) with \(p>2\). By constructing appropriate Lyapunov functionals and using LaSalle's invariant principle, we further prove that the solution of (*) converges to the co-existence steady state exponentially or competitive exclusion steady state algebraically as time tends to infinity in different parameter regimes. Our results reveal that once the resource \(w\) has temporal dynamics, two competitors may coexist in the case of weak competition regardless of their dispersal rates and initial values no matter whether there is explicit dependence in dispersal or not. When the prey's resource is spatially heterogeneous (i.e. \(m(x)\) is non-constant), we use numerical simulations to demonstrate that the striking phenomenon ``slower diffuser always prevails'' (cf. [\textit{J. Dockery} et al., ibid. 37, No. 1, 61--83 (1998; Zbl 0921.92021);\textit{Y. Lou}, J. Differ. Equations 223, No. 2, 400--426 (2006; Zbl 1097.35079)]) fails to appear if the non-random dispersal strategy is employed by competing species (i.e. either \(d_1(w)\) or \(d_2(w)\) is non-constant) while it still holds true if both \(d(w)\) and \(d_2(w)\) are constant.Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations.https://www.zbmath.org/1456.351942021-04-16T16:22:00+00:00"Waurick, Marcus"https://www.zbmath.org/authors/?q=ai:waurick.marcusSummary: The continuous dependence of solutions to certain equations on the coefficients is addressed. The class of equations under consideration has only recently be shown to be well posed. We give criteria that guarantee that convergence of the coefficients in the weak operator topology implies weak convergence of the respective solutions. We discuss three examples: A homogenization problem for a Kelvin-Voigt model for elasticity, the discussion of continuous dependence of the coefficients for acoustic waves with impedance type boundary conditions and a singular perturbation problem for a mixed type equation. By means of counterexamples, we show optimality of the results obtained.
For the entire collection see [Zbl 1420.78002].Turing conditions for pattern forming systems on evolving manifolds.https://www.zbmath.org/1456.350292021-04-16T16:22:00+00:00"Van Gorder, Robert A."https://www.zbmath.org/authors/?q=ai:van-gorder.robert-a"Klika, Václav"https://www.zbmath.org/authors/?q=ai:klika.vaclav"Krause, Andrew L."https://www.zbmath.org/authors/?q=ai:krause.andrew-lSummary: The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.Persistence of the steady normal shock structure for the unsteady potential flow.https://www.zbmath.org/1456.351372021-04-16T16:22:00+00:00"Fang, Beixiang"https://www.zbmath.org/authors/?q=ai:fang.beixiang"Xiang, Wei"https://www.zbmath.org/authors/?q=ai:xiang.wei"Xiao, Feng"https://www.zbmath.org/authors/?q=ai:xiao.fengThe authors consider unsteady potential flow in two spatial dimensions in a nozzle. No symmetry is assumed. The authors focus on shock structures and their stability with respect to perturbations of the nozzle and the initial data. The shock structure is proved to persist, at least for a short time, under perturbations of the nozzle boundary as well as of the initial data of the flow fields.
Reviewer: Ilya A. Chernov (Petrozavodsk)Amplitude blowup in radial isentropic Euler flow.https://www.zbmath.org/1456.350462021-04-16T16:22:00+00:00"Jenssen, Helge Kristian"https://www.zbmath.org/authors/?q=ai:jenssen.helge-kristian"Tsikkou, Charis"https://www.zbmath.org/authors/?q=ai:tsikkou.charisThe authors consider inviscid isentropic flow described by the compressible Euler system in 2 or 3 spatial dimensions. An additional assumption of symmetry is made: velocity is radial, and all variables depend on the distance to the origin. The aim is to prove the possibility of unbounded solutions due to wave focusing possible in several spatial dimensions.
Reviewer: Ilya A. Chernov (Petrozavodsk)Time harmonic wave propagation in one dimensional weakly randomly perturbed periodic media.https://www.zbmath.org/1456.352412021-04-16T16:22:00+00:00"Fliss, Sonia"https://www.zbmath.org/authors/?q=ai:fliss.sonia"Giovangigli, Laure"https://www.zbmath.org/authors/?q=ai:giovangigli.laureSummary: In this work we consider the solution of the time harmonic wave equation in a one dimensional periodic medium with weak random perturbations. More precisely, we study two types of weak perturbations: (1) the case of stationary, ergodic and oscillating coefficients, the typical size of the oscillations being small compared to the wavelength and (2) the case of rare random perturbations of the medium, where each period has a small probability to have its coefficients modified, independently of the other periods. Our goal is to derive an asymptotic approximation of the solution with respect to the small parameter. This can be used in order to construct absorbing boundary conditions for such media.Weakly nonlinear wave interactions in multi-degree of freedom periodic structures.https://www.zbmath.org/1456.740282021-04-16T16:22:00+00:00"Manktelow, Kevin L."https://www.zbmath.org/authors/?q=ai:manktelow.kevin-l"Leamy, Michael J."https://www.zbmath.org/authors/?q=ai:leamy.michael-j"Ruzzene, Massimo"https://www.zbmath.org/authors/?q=ai:ruzzene.massimoSummary: This work presents a multiple time scales perturbation analysis for analyzing weakly nonlinear wave interactions in multi-degree of freedom periodic structures. The perturbation analysis is broadly applicable to (discretized) periodic systems in any dimensional space and with a wide range of constitutive nonlinearities. Specific emphasis is placed on cubic nonlinearity, as dispersion shifts typically arise from the cubic components in nonlinear restoring forces. The procedure is first presented in general. Then, application to the diatomic chain and monoatomic two-dimensional lattice demonstrates, individually, the treatment of multiple degree of freedom systems and higher dimensional spaces. The dispersion relations are modified by weakly nonlinear wave interactions and lead to additional opportunities to control wave propagation direction, band gap size, and group velocity. Numerical simulations validate the expected dispersion shifts. An amplitude-tunable focus device demonstrates the viability of utilizing dynamically-introduced dispersion to produce beam steering that may, ultimately, lead to a phononic superprism effect as well as multiplexing/demultiplexing behavior.Nonlinear interfacial waves in a circular cylindrical container subjected to a vertical excitation.https://www.zbmath.org/1456.760352021-04-16T16:22:00+00:00"Chang, L."https://www.zbmath.org/authors/?q=ai:chang.lu|chang.lidan|chang.lianli|chang.leran|chang.liz|chang.long|chang.lijun|chang.li|chang.liwei|chang.linyan|chang.luping|chang.liu|chang.liping|chang.lihong|chang.lisheng|chang.linlin|chang.lizhen|chang.ling|chang.liwu|chang.leilei|chang.luo|chang.liang|chang.lee|chang.lungyi|chang.lena|chang.liangming|chang.lifang|chang.lei|chang.lili|chang.liezhen|chang.lubin|chang.lina|chang.le"Jian, Y. J."https://www.zbmath.org/authors/?q=ai:jiang.yanjie|jian.yongjun"Su, J."https://www.zbmath.org/authors/?q=ai:su.jin|su.jianmin|su.juanli|su.jessica|su.jinxia|su.juntong|su.jingrui|su.jianfeng|su.jiexian|su.jianxin|su.jincheng|su.jindian|su.jianyong|su.jinya|su.jia|su.jianji|su.jinling|su.jianbo|su.jiaduo|su.jinshu|su.jiuqing|su.jiawei|su.jionglong|su.jianwen|su.jiang|su.jingyong|su.jianning|su.jichao|su.junwei|su.julie|su.jingguo|su.jianbin|su.junshan|su.junchen|su.juguo|su.jingjing|su.jifeng|su.jonathan|su.jianxiu|su.jinyong|su.jie|su.jianbing|su.juan|su.jingxun|su.jian|su.jingbo|su.jing|su.jianxi|su.jun|su.jiabao|su.jianzhong|su.jihong|su.jianhua|su.jinxuan|su.jiafu|su.jianye"Na, R."https://www.zbmath.org/authors/?q=ai:na.risa|na.renmandula"Liu, Q. S."https://www.zbmath.org/authors/?q=ai:liu.quansen|liu.qingshan|liu.qingsong|liu.qinsheng|liu.quanshan|liu.qiangsheng|liu.quansheng|liu.qiongsun|liu.quansheng.1|liu.qiusheng|liu.qingsheng|liu.qiushan|liu.quanshen|liu.qunsheng|liu.qi-shan"He, G. W."https://www.zbmath.org/authors/?q=ai:he.guowei|he.guangweiSummary: Singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single interface standing wave in a two-layer liquid-filled circular cylindrical vessel, which is subjected to a vertical periodical oscillation. It is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear amplitude equation including cubic nonlinear and vertically forced terms, was derived by the method of expansion of two-time scales without taking the influence of surface tension into account. By numerical computation, it is shown that different patterns of interface standing wave can be excited for different driving frequency and amplitude. We found that the interface wave mode become more and more complex as increasing of upper to lower layer density ratio \(\gamma\). The traits of the standing interface wave were proved theoretically. In addition, the dispersion relation and nonlinear amplitude equation obtained in this article can reduce to the known results for a single fluid when \(\gamma = 0\), \(h_2 \rightarrow h_1\).Spectral stability of nonlinear waves in KdV-type evolution equations.https://www.zbmath.org/1456.370792021-04-16T16:22:00+00:00"Pelinovsky, Dmitry E."https://www.zbmath.org/authors/?q=ai:pelinovsky.dmitry-eAuthor's abstract: This chapter focuses on the spectral stability of nonlinear waves in Korteweg-de Vries (KdV) type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a finite number of negative eigenvalues and an unbounded non-invertible operator \(\partial_x\). The instability index theorem is proven under a generic assumption on the self-adjoint operator both in the case of solitary waves and periodic waves. This result is reviewed in the context of recent results on spectral stability of nonlinear waves in KdV-type evolution equations.
For the entire collection see [Zbl 1280.37002].
Reviewer: Kaïs Ammari (Monastir)A posteriori modeling error estimates in the optimization of two-scale elastic composite materials.https://www.zbmath.org/1456.651592021-04-16T16:22:00+00:00"Conti, Sergio"https://www.zbmath.org/authors/?q=ai:conti.sergio"Geihe, Benedict"https://www.zbmath.org/authors/?q=ai:geihe.benedict"Lenz, Martin"https://www.zbmath.org/authors/?q=ai:lenz.martin"Rumpf, Martin"https://www.zbmath.org/authors/?q=ai:rumpf.martinSummary: The a posteriori analysis of the discretization error and the modeling error is studied for a compliance cost functional in the context of the optimization of composite elastic materials and a two-scale linearized elasticity model. A mechanically simple, parametrized microscopic supporting structure is chosen and the parameters describing the structure are determined minimizing the compliance objective. An a posteriori error estimate is derived which includes the modeling error caused by the replacement of a nested laminate microstructure by this considerably simpler microstructure. Indeed, nested laminates are known to realize the minimal compliance and provide a benchmark for the quality of the microstructures. To estimate the local difference in the compliance functional the dual weighted residual approach is used. Different numerical experiments show that the resulting adaptive scheme leads to simple parametrized microscopic supporting structures that can compete with the optimal nested laminate construction. The derived a posteriori error indicators allow to verify that the suggested simplified microstructures achieve the optimal value of the compliance up to a few percent. Furthermore, it is shown how discretization error and modeling error can be balanced by choosing an optimal level of grid refinement. Our two scale results with a single scale microstructure can provide guidance towards the design of a producible macroscopic fine scale pattern.Stability and existence of stationary solutions to the Euler-Poisson equations in a domain with a curved boundary.https://www.zbmath.org/1456.828882021-04-16T16:22:00+00:00"Suzuki, Masahiro"https://www.zbmath.org/authors/?q=ai:suzuki.masahiro"Takayama, Masahiro"https://www.zbmath.org/authors/?q=ai:takayama.masahiroSummary: The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of walls immersed in a plasma, and to analyze qualitative information of such a sheath layer. In the case of planar wall, Bohm proposed a criterion on the velocity of the positive ion for the formation of sheath, and several works gave its mathematical validation. It is of more interest to analyze the criterion for the nonplanar wall. In this paper, we study the existence and asymptotic stability of stationary solutions for the Euler-Poisson equations in a domain of which boundary is drawn by a graph. The existence and stability theorems are shown by assuming that the velocity of the positive ion satisfies the Bohm criterion at infinite distance. What most interests us in these theorems is that the criterion together with a suitable necessary condition guarantees the formation of sheaths as long as the shape of walls is drawn by a graph.Fluctuations around a homogenised semilinear random PDE.https://www.zbmath.org/1456.352432021-04-16T16:22:00+00:00"Hairer, Martin"https://www.zbmath.org/authors/?q=ai:hairer.martin"Pardoux, Étienne"https://www.zbmath.org/authors/?q=ai:pardoux.etienneSummary: We consider a semilinear parabolic partial differential equation in \(\mathbb{R}_+ \times [0, 1]^d\), where \(d = 1, 2\) or 3, with a highly oscillating random potential and either homogeneous Dirichlet or Neumann boundary condition. If the amplitude of the oscillations has the right size compared to its typical spatiotemporal scale, then the solution of our equation converges to the solution of a deterministic homogenised parabolic PDE, which is a form of law of large numbers. Our main interest is in the associated central limit theorem. Namely, we study the limit of a properly rescaled difference between the initial random solution and its LLN limit. In dimension \(d = 1\), that rescaled difference converges as one might expect to a centred Ornstein-Uhlenbeck process. However, in dimension \(d = 2\), the limit is a non-centred Gaussian process, while in dimension \(d = 3\), before taking the CLT limit, we need to subtract at an intermediate scale the solution of a deterministic parabolic PDE, subject (in the case of Neumann boundary condition) to a non-homogeneous Neumann boundary condition. Our proofs make use of the theory of regularity structures, in particular of the very recently developed methodology allowing to treat parabolic PDEs with boundary conditions within that theory.Effective models and numerical homogenization for wave propagation in heterogeneous media on arbitrary timescales.https://www.zbmath.org/1456.350152021-04-16T16:22:00+00:00"Abdulle, Assyr"https://www.zbmath.org/authors/?q=ai:abdulle.assyr"Pouchon, Timothée"https://www.zbmath.org/authors/?q=ai:pouchon.timotheeThe authors propose a family of effective equations for wave propagation in periodic media derived for arbitrary timescales. The main effort is paid on proving quantitative error (corrector) estimates for the homogenized tensor and other macroscopic quantities. In particular, they discover a relation between the correctors of arbitrary order, which allows to reduce the computational cost of the effective tensors.
Reviewer: Adrian Muntean (Karlstad)Viscosity robust weak Galerkin finite element methods for Stokes problems.https://www.zbmath.org/1456.651682021-04-16T16:22:00+00:00"Wang, Bin"https://www.zbmath.org/authors/?q=ai:wang.bin.3|wang.bin.2|wang.bin.4|wang.bin|wang.bin.1"Mu, Lin"https://www.zbmath.org/authors/?q=ai:mu.linSummary: In this paper, we develop a viscosity robust weak Galerkin finite element scheme for Stokes equations. The major idea for achieving pressure-independent energy-error estimate is to use a divergence preserving velocity reconstruction operator in the discretization of the right hand side body force. The optimal convergence results for velocity and pressure have been established in this paper. Finally, numerical examples are presented for validating the theoretical conclusions.A microscopic model for a one parameter class of fractional Laplacians with Dirichlet boundary conditions.https://www.zbmath.org/1456.352102021-04-16T16:22:00+00:00"Bernardin, C."https://www.zbmath.org/authors/?q=ai:bernardin.cedric"Gonçalves, P."https://www.zbmath.org/authors/?q=ai:goncalves.patricia-c"Jiménez-Oviedo, B."https://www.zbmath.org/authors/?q=ai:jimenez-oviedo.bSummary: We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density \(\alpha\) at the left of the system and \(\beta\) at the right of the system. The strength of the reservoirs is ruled by \(\kappa N^{-\theta} > 0\). Here \(N\) is the size of the system, \(\kappa > 0\) and \(\theta \in \mathbb{R}\). Our results are valid for \(\theta \le 0\). For \(\theta = 0\), we obtain a collection of fractional reaction-diffusion equations indexed by the parameter \(\kappa\) and with Dirichlet boundary conditions. Their solutions also depend on \(\kappa\). For \(\theta < 0\), the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case \(\theta > 0\) is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case \(\theta = 0\) when we send the parameter \(\kappa\) to zero. Indeed, we conjecture that the limiting profile when \(\kappa \rightarrow 0\) is the one that we should obtain when taking small values of \(\theta > 0\).Resonances of three transverse standing-wave modes in a waveguide with periodic wall undulations.https://www.zbmath.org/1456.741002021-04-16T16:22:00+00:00"Tao, Zhi-Yong"https://www.zbmath.org/authors/?q=ai:tao.zhiyong"Fan, Ya-Xian"https://www.zbmath.org/authors/?q=ai:fan.ya-xianSummary: Interactions of three transverse modes are investigated in a waveguide with periodic walls. Resonances of two guided wave modes always result in forbidden bands for wave propagations when the wavenumber matching conditions are satisfied. As a third mode is involved due to the selected wall corrugations, we find that a single high-order mode can penetrate through the forbidden band based on the complex interactions. A method for generating a single high-order transverse mode is proposed by manipulating the multimode interactions. The numerical simulations on acoustic waveguides, showing the extreme suppression of the unwanted modes in the Bragg and non-Bragg gaps, demonstrate the validity and the efficiency of the proposed method.A proof via finite elements for Schiffer's conjecture on a regular pentagon.https://www.zbmath.org/1456.651652021-04-16T16:22:00+00:00"Nigam, Nilima"https://www.zbmath.org/authors/?q=ai:nigam.nilima-a|nigam.nilima"Siudeja, Bartłomiej"https://www.zbmath.org/authors/?q=ai:siudeja.bartlomiej-andrzej"Young, Benjamin"https://www.zbmath.org/authors/?q=ai:young.benjamin-jThe authors show that there exists a Neumann eigenfunction on a regular pentagon that is positive on the boundary and not identically constant. The proof of this result consists in blending ideas from analysis, optimization and numerical analysis (approximation theory techniques), and it is an interesting contribution in its own right. The proof itself is human-readable and fully analytic except for the matrix eigenvalue computations on 20 large matrices from the strategy for computing discrete eigenvalues and from the lower bound estimation for the smallest eigenvalue of a sparse matrix. The excepted part is of course a computed assisted proof. The authors have found a rather short proof certificate for this computed proof, show that it is checkable and adaptable.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain.https://www.zbmath.org/1456.350432021-04-16T16:22:00+00:00"Yang, Shuang"https://www.zbmath.org/authors/?q=ai:yang.shuang"Li, Yangrong"https://www.zbmath.org/authors/?q=ai:li.yangrongSummary: A pullback random attractor is called forward controllable if its time-component is semi-continuous to a compact set in the future, and the minimum among all such compact limit-sets is called a forward controller. The existence of a forward controller closely relates to the forward compactness of the attractor, which is further argued by the forward-pullback asymptotic compactness of the system. The abstract results are applied to the non-autonomous stochastic sine-Gordon equation on an unbounded domain. The existence of a forward compact attractor is proved, which leads to the existence of a forward controller. The measurability of the attractor is proved by considering two different universes.Global solutions and large time behavior for the chemotaxis-shallow water system.https://www.zbmath.org/1456.351702021-04-16T16:22:00+00:00"Zhai, Xiaoping"https://www.zbmath.org/authors/?q=ai:zhai.xiaoping"Chen, Yiren"https://www.zbmath.org/authors/?q=ai:chen.yirenThe authors study a chemotaxis system coupled with the viscous shallow water equations which models bacteriae chemotactic movement towards higher concentration of oxygen they consume, under gravitational potential and convective transport effects.
A well-posedness result is proved for that two-dimensional evolution problem with small initial data in the critical Besov spaces. Time decay of solutions -- coinciding with that for the diffusion asymptotics -- is proved.
Reviewer: Piotr Biler (Wrocław)Global conservative solutions for a modified periodic coupled Camassa-Holm system.https://www.zbmath.org/1456.350572021-04-16T16:22:00+00:00"Chen, Rong"https://www.zbmath.org/authors/?q=ai:chen.rong"Pan, Shihang"https://www.zbmath.org/authors/?q=ai:pan.shihang"Zhang, Baoshuai"https://www.zbmath.org/authors/?q=ai:zhang.baoshuaiSummary: In present paper, we deal with the behavior of a solution beyond the occurrence of wave breaking for a modified periodic coupled Camassa-Holm system. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, this evolution system is rewritten as a closed semilinear system. The local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows us to continue the solution after wave breaking, and gives a global conservative solution where the energy is conserved for almost all times. Returning to the original variables. We finally obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Additionally, our results repair some gaps in the pervious work.The scaling hypothesis for Smoluchowski's coagulation equation with bounded perturbations of the constant kernel.https://www.zbmath.org/1456.350652021-04-16T16:22:00+00:00"Cañizo, José A."https://www.zbmath.org/authors/?q=ai:canizo.jose-alfredo"Throm, Sebastian"https://www.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
\[
\partial_\tau \phi(\tau,\xi) = \frac{1}{2} \int_0^\xi K(\xi-\eta,\eta) \phi(\tau,\xi-\eta) \phi(\tau,\eta)\ d\eta
- \int_0^\infty K(\xi,\eta) \phi(\tau,\xi) \phi(\tau,\eta)\ d\eta
\]
where \((\tau,\xi)\in (0,\infty)\times (0,\infty)\), with non-negative initial condition \(\phi_0\in L^1((0,\infty),\xi d\xi)\), behave in a self-similar way for large values of \(\tau\). This conjecture is up to now known to be true for the \textit{so-called} solvable kernels \(K(\xi,\eta)=2\) and \(K(\xi,\eta)=\xi+\eta\), see [\textit{G. Menon} and \textit{R. L. Pego}, Commun. Pure Appl. Math. 57, No. 9, 1197--1232 (2004; Zbl 1049.35048)]. Its validity is extended here to small perturbations of the constant kernel with homogeneity zero. In addition, a temporal decay rate is derived. More precisely, let \(W\in C((0,\infty)^2)\) be a symmetric function satisfying
\[
0 \le W(\xi,\eta) \leq 1\text{ and } W(\lambda\xi,\lambda\eta) = W(\xi,\eta), \qquad (\lambda,\xi,\eta)^3,
\]
and set \(K_\varepsilon = 2 + \varepsilon W\) for \(\varepsilon \ge 0\). It is shown that, for \(\varepsilon>0\) sufficiently small, there is a unique self-similar solution \((\tau,\xi) \mapsto (1+\tau)^{-2} G_\varepsilon(x(1+\tau)^{-1})\) such that \(\|G_\varepsilon\|_{L^1((0,\infty),\xi d\xi)}=1\) and \(G_\varepsilon\in L^1((0,\infty),\xi^k d\xi)\) for all \(k\ge 0\). It is further proved that this self-similar solution is stable in the following sense: given \(R>0\), \(k>2\), and a non-negative initial condition \(\phi_0\) satisfying
\[
\int_0^\infty \xi \phi_0(\xi)\ \mathrm{d}\xi = 1\,, \quad \int_0^\infty |\phi_0(\xi) -G_\varepsilon(\xi)| (1+\xi)^k\ \mathrm{d}\xi \le R\,,
\]
there are \(M>0\) and \(C>0\) depending only on \(R\) and \(k\) such that
\begin{align*}
& \int_0^\infty |(1+\tau)^2 \phi(\tau,x (1+\tau)) - G_\varepsilon(x)| (1+x)^k\ \mathrm{d}x \cr
& \qquad\qquad \le C (1+\tau)^{(1-2M\varepsilon)/2} \int_0^\infty |\phi_0(\xi) -G_\varepsilon(\xi)| (1+\xi)^k\ \mathrm{d}\xi
\end{align*}
for \(\tau\ge 0\). The proof relies on a refined study of the dynamics of the coagulation equation with constant kernel \(K_0\), building upon previous works on this particular case. In particular, a spectral gap for the linearised operator around the explicit self-similar profile \(G_0(x) = e^{-x}\) is obtained. Also, the stability of the self-similar profiles \((G_\varepsilon)\) with respect to \(\varepsilon\) is established.
Reviewer: Philippe Laurençot (Toulouse)Global analysis of quasilinear wave equations on asymptotically de Sitter spaces.https://www.zbmath.org/1456.351412021-04-16T16:22:00+00:00"Hintz, Peter"https://www.zbmath.org/authors/?q=ai:hintz.peterSummary: We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces. We obtain our results by proving the global invertibility of linear operators with coefficients in high regularity \(L^2\)-based function spaces and using iterative arguments for the nonlinear problems. The linear analysis is accomplished in two parts: Firstly, a regularity theory is developed by means of a calculus for pseudodifferential operators with non-smooth coefficients, similar to the one developed by Beals and Reed, on manifolds with boundary. Secondly, the asymptotic behavior of solutions to linear equations is studied using resonance expansions, introduced in this context by Vasy using the framework of Melrose's \(b\)-analysis.The streamline-diffusion finite element method on graded meshes for a convection-diffusion problem.https://www.zbmath.org/1456.651692021-04-16T16:22:00+00:00"Yin, Yunhui"https://www.zbmath.org/authors/?q=ai:yin.yunhui"Zhu, Peng"https://www.zbmath.org/authors/?q=ai:zhu.pengSummary: In this paper, the streamline-diffusion finite element method is applied to a two-dimensional convection-diffusion problem posed on the unit square, using a graded mesh of \(\mathcal{O}(N^2)\) points based on standard Lagrange polynomials of degree \(k \geq 1\). We prove the method is convergent almost uniformly in the perturbation parameter \(\in\), and a convergence order \(\mathcal{O}\left(N^{- k} \log^{k + 1}(\frac{1}{\epsilon})\right)\) is obtained in a streamline-diffusion norm under certain assumptions. Numerical experiments support the theoretical results.Dynamics and stability of sessile drops with contact points.https://www.zbmath.org/1456.351542021-04-16T16:22:00+00:00"Tice, Ian"https://www.zbmath.org/authors/?q=ai:tice.ian"Wu, Lei"https://www.zbmath.org/authors/?q=ai:wu.lei.1Authors' abstract: The authors consider the dynamics of a two-dimensional droplet of incompressible viscous fluid evolving above a one-dimensional flat surface under the influence of gravity. This is a free boundary problem: the interface between the fluid on the surface and the air above is free to move and experience capillary forces. A mathematical model of this problem is formulated and some a priori estimates are obtained. These estimates are used to show that for initial data sufficiently close to equilibrium, there exist global solutions of the model that decay to a shifted equilibrium exponentially fast.
Reviewer: Gheorghe Moroşanu (Cluj-Napoca)Existence, multiplicity and regularity of solutions for the fractional \(p\)-Laplacian equation.https://www.zbmath.org/1456.352162021-04-16T16:22:00+00:00"Kim, Yun-Ho"https://www.zbmath.org/authors/?q=ai:kim.yunho.1|kim.yunhoSummary: : We are concerned with the following elliptic equations:
\[
\begin{cases} (-\Delta)_p^su=\lambda f(x,u) \quad \text{in }\Omega\\u= 0\quad\text{on }\mathbb{R}^N\backslash\Omega,\end{cases}
\]
where \(\lambda\) are real parameters, \((-\Delta)_p^s\) is the fractional \(p\)-Laplacian operator, \(0 < s < 1 < p < +\infty, sp < N\), and \(f:\Omega\times\mathbb R\to\mathbb R\) satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters \(\lambda\) for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity \(f\) has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in \(L^{\infty}(\Omega)\) of any possible weak solution by applying the bootstrap argument.