Quasilinear hyperbolic systems with involutions
type: Item
instance of: scholarly article
title: Quasilinear hyperbolic systems with involutions
zbMATH Open document ID: 0614.35057
DOI: 10.1007/BF00280911
author: Constantine M. Dafermos
published in: Archive for Rational Mechanics and Analysis
publication date: 1986-01-01T00:00:00Z
review text: The author considers quasilinear hyperbolic systems \[ (1)\quad \partial_ tU+\sum^{m}_{\alpha =1}\partial_{\alpha}G_{\alpha}(U)=0 \] where \(x\in {\mathbb{R}}^ m\), the vector U(x,t) takes values in an open subset \({\mathcal O}\subset {\mathbb{R}}^ n\) and \(G_{\alpha}: {\mathcal O}\to {\mathbb{R}}^ n\) are given smooth functions. A classical solution of (1) is a uniformly Lipschitz continuous function \(U=U({\mathbb{R}}^ m\times [0,\tau))\) which satisfies (1) almost everywhere. A weak solution of the class of functions with bounded variation (BV) is a bounded measurable function \(U({\mathbb{R}}^ m\times [0,\tau))\) with distributional derivatives \(\partial_ tU\), \(\partial_{\alpha}U\) which satisfies (1) in the sense of distribution. It can be shown that the Cauchy problem for (1) may have several solutions of class BV even after imposing the additional requirement that they satisfy an entropy inequality \[ (2)\quad \partial_ t\eta (U)+\sum^{m}_{\alpha =1}\partial_{\alpha}q_{\alpha}(U)\leq 0 \] with \(\eta\) convex (\(\eta\) entropy, q entropy flux). The author shows, whenever a classical solution exists the entropy inequality (2) with \(\eta\) strictly K-convex (see Def. 2.1 in the paper) manages to rule out all other weak solutions of class BV with shocks of moderate strength.
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 3994129
zbMATH Keywords: quasilinear hyperbolic systems
zbMATH Keywords: classical solution
zbMATH Keywords: weak solution
zbMATH Keywords: Cauchy problem
zbMATH Keywords: entropy inequality
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00280911
OpenAlex ID: W4231817347
description: scientific article; zbMATH DE number 3994129