Singular kernels and compactness in nonlinear conservation laws
type: Item
instance of: scholarly article
title: Singular kernels and compactness in nonlinear conservation laws
zbMATH Open document ID: 0911.35071
author: Hana Petzeltová
author: Eduard Feireisl
published in: Journal of Differential Equations
publication date: 1999-05-18T00:00:00Z
review text: A nonlinear equation of the form \[ {\partial\over\partial t}(K*u)+\sum^N_{i=1}a_i(u){\partial u\over\partial x_i}=0,\qquad u=u(t,x),\quad t\in(0,T),\;x\in\mathbb{R}^N \] where \(K*u(t,x)\equiv\int^t_{-\infty}K(t-s)u(s,x)ds\), and it is supposed \(K\in L^1(\mathbb{R}^+)\); \(K\) nonnegative, nonincreasing and convex on \(\mathbb{R}^+\); \(0<k_1\leq\lim_{t\to 0_+}\inf t^\alpha K(t)\leq\lim_{t\to 0_+}\sup t^\alpha K(t)\leq k_2<\infty\), where \(\alpha\in (0,1)\) and \(k_1,k_2\) are suitable constants; \(a_i\in C^2(\mathbb{R}^1)\), meas \(\{v\in[-Y,Y]\left|\sum^N_{i=1}a_i(v)\xi_i\right.=0\}=0\) for all \(|\xi|\neq 0\). The authors proved that the set of all entropy solutions \(u\) satisfying \(-Y\leq u(t,x)\leq Y\) for a.e. \(t\in(-\infty,T)\), \(x\in\mathbb{R}^N\) is compact in \(L^1_{\text{loc}}([0,T]\times\mathbb{R}^N)\). This compactness result may be considered as a generalization of the velocity average lemmata of Di Perna, Lions and Meyer to anisotropic problems containing fractional derivatives.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L60
zbMATH DE Number: 1212875
zbMATH Keywords: entropy solution
zbMATH Keywords: velocity average lemmata
reviewed by: Vasile Iftode
MaRDI profile type: Publication
Wikidata QID: Q59316798
full work available at URL: jdeq.1997.3361
OpenAlex ID: W2055662379
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DOI: 10.1006/JDEQ.1997.3361
description: scientific article; zbMATH DE number 1212875