Convergence of the viscosity method for a nonstrictly hyperbolic conservation law
type: Item
instance of: scholarly article
title: Convergence of the viscosity method for a nonstrictly hyperbolic conservation law
zbMATH Open document ID: 0788.35089
DOI: 10.1007/BF02096565
published in: Communications in Mathematical Physics
publication date: 1993-05-16T00:00:00Z
review text: The author considers the existence of global weak solutions for nonlinear hyperbolic conservation laws \[ v_ t+(vu)_ x=0,\quad u_ t+\left( {1\over 2} u^ 2+ \int^ v s(s+\delta)^{r-3}ds \right)_ x=0 \tag{1} \] with initial data (2) \((v(x,0)\), \(u(x,0))=(v_ 0(x)\), \(u_ 0(x))\), where \(\delta,r\) are positive constants and \(r>3\). When \(\delta=0\), (1) is motivated by the isentropic equation of gas dynamics for a polytropic gas. The global weak solutions of which had been solved for the case of \(1<r<3\) by using the Glimm difference scheme by DiPerna. In the present paper system (1) is studied with bounded measurable initial data (2) by using the established technique of compensated compactness. Through an analysis of progressing entropy waves, we establish a convergence theorem for the method of artificial viscosity applied to the system (1) and obtain the existence of the global weak solutions for the Cauchy problem (1), (2).
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 76N15
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 165645
zbMATH Keywords: viscosity solutions
zbMATH Keywords: positively invariant regions
zbMATH Keywords: entropy-entropy flux of Lax type
zbMATH Keywords: Dirac measures
zbMATH Keywords: existence of global weak solutions
zbMATH Keywords: nonlinear hyperbolic conservation laws
zbMATH Keywords: isentropic equation of gas dynamics for a polytropic gas
zbMATH Keywords: bounded measurable initial data
zbMATH Keywords: compensated compactness
zbMATH Keywords: progressing entropy waves
zbMATH Keywords: convergence theorem
zbMATH Keywords: method of artificial viscosity
zbMATH Keywords: Cauchy problem
author: Yun-Guang Lu
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02096565
OpenAlex ID: W1966617242
description: scientific article; zbMATH DE number 165645