Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics
type: Item
instance of: scholarly article
title: Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics
zbMATH Open document ID: 1029.76045
author: Hermano Frid
author: Gui-Qiang Chen
author: Yachun Li
published in: Communications in Mathematical Physics
publication date: 2002-12-01T00:00:00Z
review text: The authors study a system of conservation laws for Euler system of gas dynamics in Lagrangian coordinates \(\partial_t u+\partial_xp=0\), \(\partial_t u+\partial_xu=0\), \(\partial_t (e+u^2/2)+\partial_x(pu)=0\), where \(u,p,v\) and \(e\) are velocity, pressure, specific volume (\(v=1/\rho \), \(\rho \) density) and internal energy of the fluid, respectively. The Riemann problem for this system is a special Cauchy problem with initial data, taking two different constant states for \(x<0\) and for \(x>0\). The solutions are known as Riemann solutions or simple waves (states). The large-time behavior of the solutions in \(L^{\infty }\cap BV_{\text{loc}}(\mathbb{R}_+^2)\) is of interest here. The authors study the asymptotic stability of classical Riemann solution \(R(x/t)\) for the considered Riemann problem. The uniqueness of Riemann solution under entropy condition (entropy solution) in the class \(L^{\infty }\cap BV\) with arbitrarily large oscillation is proved. For this purpose the authors use the global behavior of shock curves in phase space, and the singularity of centered rarefaction waves near the center in physical plane. The uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large \(L^{1}\cap L^{\infty }\cap BV\) perturbation of Riemann initial data, as long as the corresponding solutions belongs to \(L^{\infty }\) and have local bounded total variation satisfying a natural condition for its growth with time.
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 1836928
zbMATH Keywords: Riemann solutions
zbMATH Keywords: entropy solution
zbMATH Keywords: conservation laws
zbMATH Keywords: Euler system
zbMATH Keywords: Lagrangian coordinates
zbMATH Keywords: gas dynamics
zbMATH Keywords: shock curve
zbMATH Keywords: Riemann problem
zbMATH Keywords: Cauchy problem
zbMATH Keywords: simple waves
zbMATH Keywords: asymptotic stability
zbMATH Keywords: uniqueness
zbMATH Keywords: entropy condition
zbMATH Keywords: large oscillation
zbMATH Keywords: phase space
zbMATH Keywords: singularity
zbMATH Keywords: centered rarefaction waves
zbMATH Keywords: physical plane
zbMATH Keywords: inviscid large-time stability
zbMATH Keywords: local bounded total variation
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
full work available at URL: s002200200615
OpenAlex ID: W2027932739
Wikidata QID: Q57386537
DOI: 10.1007/S002200200615
description: scientific article; zbMATH DE number 1836928