Nonclassical shocks and the Cauchy problem for nonconvex conservation laws
type: Item
instance of: scholarly article
title: Nonclassical shocks and the Cauchy problem for nonconvex conservation laws
zbMATH Open document ID: 0921.35107
published in: Journal of Differential Equations
publication date: 1999-09-16T00:00:00Z
review text: The authors consider the Cauchy problem for a conservation law with nonconvex flux-function (taken to be cubic, for simplicity) \[ \partial_tu+\partial_xu^3=0, \quad u(x,0)=u_0(x), \quad u(x,t)\in \mathbb{R},\;x\in \mathbb{R},\;t>0. \] The Riemann problem for this law can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality of the form \[ \partial_tU(u)+\partial_xF(u)\leq 0, \] where \(U\) is strictly convex, and \(F'(u)=3U'(u)u^2\), and kinetic relation, which is required for nonclassical shocks only and determines them uniquely. The limiting solution \(u=\lim_{\varepsilon\rightarrow 0}u_\varepsilon\) given by \[ \partial_tu_\varepsilon+\partial_xu^3_\varepsilon= \varepsilon\partial_{xx}u_\varepsilon+A\varepsilon^2\partial_{xxx}u_\varepsilon \] is referred to as the admissible nonclassical solution. The authors use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in total variation norm. The nonclassical shocks induce an increase of the total variation and, therefore, the classical measure of total variation must be modified accordingly. The authors prove that the front tracking scheme converges strongly to a weak solution satisfying the entropy inequality.
reviewed by: S. B. Medvedev
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 76L05
zbMATH DE Number: 1271352
zbMATH Keywords: nonclassical Riemann solver
zbMATH Keywords: weak solution
zbMATH Keywords: entropy
zbMATH Keywords: nonclassical shock
zbMATH Keywords: front tracking algorithm
author: Debora Amadori
author: Paolo Baiti
author: Philippe G. LeFloch
author: B. Piccoli
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3513
OpenAlex ID: W2051227185
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DOI: 10.1006/JDEQ.1998.3513
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description: scientific article; zbMATH DE number 1271352