Existence theory for nonclassical entropy solutions of scalar conservation laws
type: Item
instance of: scholarly article
title: Existence theory for nonclassical entropy solutions of scalar conservation laws
zbMATH Open document ID: 1082.35103
author: Paolo Baiti
author: Philippe G. LeFloch
author: B. Piccoli
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2005-02-24T00:00:00Z
review text: A general existence theory for the Cauchy problem for a one space dimensional conservation law: \[ \partial_ t u + \partial_ x f(u) =0, \quad u\in \mathbb R^ N,\;x\in \mathbb R,\;t>0\,, \] is presented. The flux function \(f\) is assumed to be nonconvex in this paper. The aim is to establish existence of solutions having the so-called nonclassical shocks. Notice that the classical entropy solution satisfies the Oleinik entropy inequalities and the total variation \(TV(u(.,t))\) satisfies: \(TV(u(.,t))\leq TV(u(.,0))\). The main difficulty for the solutions having nonclassical shocks lies in the fact that their total variation may increase. In this paper the authors introduce and investigate a generalized total variation functional \(V=V(u)\) for which \(V(u(.,t))\leq V(u(.,0))\) even when nonclassical shocks are involved.
reviewed by: Daniel Ševčovič
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 2138488
zbMATH Keywords: nonclassical shock
zbMATH Keywords: generalized total variation
zbMATH Keywords: one space dimension
zbMATH Keywords: entropy solution
zbMATH Keywords: wave front tracking
MaRDI profile type: Publication
full work available at URL: s00033-004-2055-x
OpenAlex ID: W4298232918
DOI: 10.1007/S00033-004-2055-X
description: scientific article; zbMATH DE number 2138488