Recent zbMATH articles in MSC 35Fhttps://www.zbmath.org/atom/cc/35F2021-04-16T16:22:00+00:00WerkzeugStream functions for divergence-free vector fields.https://www.zbmath.org/1456.350812021-04-16T16:22:00+00:00"Kelliher, James P."https://www.zbmath.org/authors/?q=ai:kelliher.james-pSummary: In 1990, von Wahl and, independently, Borchers and Sohr showed that a divergence-free vector field \(u\) in a 3D bounded domain that is tangential to the boundary can be written as the curl of a vector field vanishing on the boundary of the domain. We extend this result to higher dimension and to Lipschitz boundaries in a form suitable for integration in flat space, showing that \(u\) can be written as the divergence of an antisymmetric matrix field. We also demonstrate how obtaining a kernel for such a matrix field is dual to obtaining a Biot-Savart kernel for the domain.Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions.https://www.zbmath.org/1456.350832021-04-16T16:22:00+00:00"Kang, Hao"https://www.zbmath.org/authors/?q=ai:kang.hao"Ruan, Shigui"https://www.zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper, we develop some basic theory for age-structured population models with nonlocal diffusion and nonlocal boundary conditions. We first apply the theory of integrated semigroups and non-densely defined operators to a linear equation, study the spectrum, and analyze the asymptotic behavior via asynchronous exponential growth. Then we consider a semilinear equation with nonlocal diffusion and nonlocal boundary condition, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii's fixed point theorem to obtain the existence of nontrivial steady states, and establish the stability of steady states. Finally we generalize these results to a nonlinear equation with nonlocal diffusion and nonlocal boundary condition.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.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].On sharp fronts and almost-sharp fronts for singular SQG.https://www.zbmath.org/1456.350842021-04-16T16:22:00+00:00"Khor, Calvin"https://www.zbmath.org/authors/?q=ai:khor.calvin"Rodrigo, José L."https://www.zbmath.org/authors/?q=ai:rodrigo.jose-luisSummary: In this paper we consider a family of active scalars with a velocity field given by \(u = \Lambda^{- 1 + \alpha} \nabla^\bot \theta\), for \(\alpha \in (0, 1)\). This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to \(\alpha = 0\). We consider the evolution of sharp fronts by studying families of almost-sharp fronts. These are smooth solutions with simple geometry in which a sharp transition in the solution occurs in a tubular neighbourhood (of size \(\delta)\). We study their evolution and that of compatible curves, and introduce the notion of a spine for which we obtain improved evolution results, gaining a full power (of \(\delta)\) compared to other compatible curves.All the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function.https://www.zbmath.org/1456.350822021-04-16T16:22:00+00:00"Fujita, Yasuhiro"https://www.zbmath.org/authors/?q=ai:fujita.yasuhiro"Hamamuki, Nao"https://www.zbmath.org/authors/?q=ai:hamamuki.nao"Yamaguchi, Norikazu"https://www.zbmath.org/authors/?q=ai:yamaguchi.norikazuSummary: We determine all the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function, which is everywhere continuous and nowhere differentiable. This result clarifies how singularities of the solution propagate along generalized characteristics. Moreover it turns out that the Takagi function still keeps the validity of the recent results in [\textit{P. Albano} et al., J. Differ. Equations 268, No. 4, 1412--1426 (2020; Zbl 1437.35153)], in which locally Lipschitz continuous initial data are handled.Hamilton-Jacobi equations for neutral-type systems: inequalities for directional derivatives of minimax solutions.https://www.zbmath.org/1456.490092021-04-16T16:22:00+00:00"Lukoyanov, Nikolai Yu."https://www.zbmath.org/authors/?q=ai:lukoyanov.nikolai-yu"Plaksin, Anton R."https://www.zbmath.org/authors/?q=ai:plaksin.anton-romanovichThe authors consider the Hamilton-Jacobi (HJ) equations for differential systems of neutral type. They continue a series of study of the dynamical system from the point of view of directional derivatives in optimization context. They give an infinitesimal criterion of the minimax (generalized) solutions for the HJ equations arising in optimal control problems and differential games, using a suitable definition and techniques of directional derivatives.
Reviewer: A. Omrane (Cayenne)An \(L^2\) to \(L^\infty\) framework for the Landau equation.https://www.zbmath.org/1456.351962021-04-16T16:22:00+00:00"Kim, Jinoh"https://www.zbmath.org/authors/?q=ai:kim.jinoh"Guo, Yan"https://www.zbmath.org/authors/?q=ai:guo.yan"Hwang, Hyung Ju"https://www.zbmath.org/authors/?q=ai:hwang.hyung-juThe authors consider the Landau equation with Coulomb potential: \(\partial
_{t}F+v\cdot \nabla _{x}F=Q(F,F)=\nabla v\cdot \int_{\mathbb{R}^{3}}\phi
(v-v^{\prime })[F(v^{\prime })\nabla _{v}F(v)-F(v)\nabla _{v}F(v^{\prime
})]dv\), posed in \((0,\infty )\times \mathbb{T}^{3}\), where \(\mathbb{T}^{3}\)
is the 3D torus, \(F(t,x,v)\geq 0\) is the spatially periodic distribution
function for particles, and \(\phi \) is the non-negative matrix defined as \(
\phi ^{ij}(v)=\{\delta _{i,j}-\frac{v_{i}v_{j}}{\left\vert v\right\vert ^{2}}
\}\left\vert v\right\vert ^{-1}\). They introduce the normalized Maxwellian \(
\mu (v)=e^{-\left\vert v\right\vert ^{2}}\)\ and writing \(F(t,x,v)=\mu
(v)+f(t,x,v)\) they observe that \(f\) satisfies \(f_{t}+v\cdot \partial
_{x}f+Lf=\Gamma (f,f)\), where \(L=-A-K\) is the linear operator with \(Af=\mu
^{-1/2}\partial _{i}\{\mu ^{1/2}\sigma ^{ij}[\partial _{j}f+v_{j}f]\}\), \(
Kf=-\mu ^{-1/2}\partial _{i}\{\mu \phi ^{ij}\ast \mu ^{1/2}[\partial
_{j}f+v_{j}f]\}\), and \(\Gamma (g,f)=\partial _{i}[\{\phi ^{ij}\ast \lbrack
\mu ^{1/2}g]\}\partial _{j}f]+\{\phi ^{ij}\ast \lbrack v_{i}\mu
^{1/2}g]\}\partial _{j}f-\partial _{i}[\{\phi ^{ij}\ast \lbrack \mu
^{1/2}\partial _{j}g]\}f]+\{\phi ^{ij}\ast \lbrack v_{i}\mu ^{1/2}\partial
_{j}g]\}f\). The initial condition \(f(0,x,v)=f_{0}(x,v)\) is added, where \(
f_{0}\) satisfies the conservation laws \(\int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}f_{0}(x,v)\sqrt{\mu }=\int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}v_{i}f_{0}(x,v)\sqrt{\mu }=\int \int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}\left\vert v\right\vert ^{2}f_{0}(x,v)\sqrt{\mu }=0\). The authors
define the notion of weak solution to this problem as a function \(
f(t,x,v)\in L^{\infty }((0,\infty )\times \mathbb{T}^{3}\times \mathbb{R}
^{3},w^{\vartheta }(v)dtdxdv)\), which satisfies \(\int_{0}^{T}\left\Vert
f(s)\right\Vert _{\sigma ,\vartheta }^{2}ds<+\infty \) and a variational
formulation issued from the above equation. Here \(\left\Vert f(s)\right\Vert
_{\sigma ,\vartheta }^{2}=\int \int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}w^{2\vartheta }[\sigma ^{ij}\partial _{i}f\partial _{j}f+\sigma
^{ij}v_{i}v_{j}f^{2}]dvdx\). The main result of the paper proves the
existence of a unique weak solution to this problem, if the initial data \(
f_{0}\) satisfies \(\left\Vert f_{0}\right\Vert _{\infty ,\vartheta }^{2}\leq
\varepsilon _{0}\) and \(\left\Vert -v\cdot \nabla _{v}f_{0}+\overline{A}
_{f_{0}}f_{0}\right\Vert _{\infty ,\vartheta }+\left\Vert
D_{v}f_{0}\right\Vert _{\infty ,\vartheta }<\infty \) for some \(\varepsilon
_{0}\in (0,1]\) and some positive \(\vartheta \). This weak solution satisfies
different estimates. For the proof, the authors first consider the
linearized Landau equation \(\partial _{t}f+v\cdot \partial _{x}f+Lf=\Gamma
(g,f)\), for some bounded function \(g\). They establish a uniform \(L^{2}\)
-estimate on a\ classical solution to the original problem and to this
linearized problem if \(\left\Vert g\right\Vert _{\infty }\) is small enough,
from which they then deduce a uniform \(L^{\infty }\)-estimate and a \(%
C^{0,\alpha }\)-estimate, through \(L^{2}-L^{\infty }\) estimates for the
solution of auxiliary linear problems. This allows deriving an Hölder
estimate and a \(S^{p}\)-estimate for the solution to the linearized problem,
where \(\left\Vert f\right\Vert _{S^{p}(\Omega )}=\left\Vert f\right\Vert _{L^{p}(\Omega )}+\left\Vert
D_{v}f\right\Vert _{L^{p}(\Omega )}+\left\Vert D_{vv}f\right\Vert
_{L^{p}(\Omega )}+\left\Vert (-\partial _{t}-v\cdot \nabla _{x})f\right\Vert
_{L^{p}(\Omega )}\), with \(\Omega =(0,\infty )\times \mathbb{T}^{3}\times
\mathbb{R}^{3}\).
Reviewer: Alain Brillard (Riedisheim)Meromorphic 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 \(\mathbb{C}^2\).