\(L^1\)-theory of scalar conservation law with continuous flux function
type: Item
instance of: scholarly article
title: \(L^1\)-theory of scalar conservation law with continuous flux function
zbMATH Open document ID: 0944.35048
author: Boris Andreianov
author: S. N. Kruzhkov
author: Philippe Benilan
published in: Journal of Functional Analysis
publication date: 2000-08-28T00:00:00Z
review text: The authors study uniqueness of a generalized entropy solution (g.e.s.) to the Cauchy problem for \(N\)-dimensional scalar conservation law \(u_t+\text{div}_xF(u)=g\), \(u(.,0)=h\), where the flux function \(F\) is only continuous. For data \((h,g)\) vanishing at infinity, it is shown that there exists a maximal and a minimal g.e.s. to the Cauchy problem and to the associated stationary problem \(u+\text{div}_xF(u) =h\). More precisely, \(h\) is supposed to be of the form \(h=c+h_0\), \(c\in\mathbb{R}\), \(h_0\in L^\infty_0(\mathbb{R}^N)\). Then the minimal and the maximal solutions coincide except for a countable set of values of \(c\) depending on \(F\), \(h_0\) and \(g\). In the case of \(L^1\) data it is proved that there is uniqueness for all data of g.e.s. to the Cauchy problem iff there is uniqueness for all data of g.e.s. to the related stationary problem. The proof uses nonlinear semigroup theory. The results are applied on the dimension \(N>1\) and extends the uniqueness results for flux having some monotonicity properties.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 1423450
zbMATH Keywords: uniqueness of a generalized entropy solution
zbMATH Keywords: nonlinear semigroup theory
reviewed by: Alexandr Doktor
MaRDI profile type: Publication
full work available at URL: jfan.1999.3445
OpenAlex ID: W1966764934
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DOI: 10.1006/JFAN.1999.3445
description: scientific article; zbMATH DE number 1423450