Weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions
type: Item
instance of: scholarly article
title: Weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions
zbMATH Open document ID: 0921.35097
author: Huijiang Zhao
published in: Journal of Differential Equations
publication date: 1999-09-16T00:00:00Z
review text: The author considers the two-dimensional scalar conservation law \[ u_t+f(u)_x+g(u)_y=0, \quad u\in \mathbb{R} \] and the corresponding relaxation system \[ u_t+v_x+w_y=0, \quad v_t+au_x+\frac{v-f(u)}{\varepsilon}=0,\quad w_t+bu_y+\frac{w-g(u)}{\varepsilon}=0. \] The weakly nonlinear limit for the relaxation approximation of the conservation law is derived through asymptotic equations and justified by employing energy estimates. Although two-dimensional results are present, and the main results hold for arbitrary space dimensions. Compared with the former works, the main difficulty lies in the problem of the existence of an (convex) entropy. To overcome this difficulty, the author obtains an estimate that is quite different from the previous estimates.
reviewed by: S. B. Medvedev
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35C20
zbMATH DE Number: 1271368
zbMATH Keywords: entropy
zbMATH Keywords: asymptotic expansion
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3498
OpenAlex ID: W2080195973
cites work: Equivalent Norms for Sobolev Spaces
cites work: Fluid dynamic limits of kinetic equations II convergence proofs for the boltzmann equation
cites work: The fluid-dynamical limit of a nonlinear model boltzmann equation
cites work: The Boltzmann equation and its applications
cites work: Hyperbolic conservation laws with stiff relaxation terms and entropy
cites work: Zero relaxation and dissipation limits for hyperbolic conservation laws
cites work: A convex entropy for a hyperbolic system with relaxation
cites work: The relaxation schemes for systems of conservation laws in arbitrary space dimensions
cites work: Hyperbolic conservation laws with relaxation
cites work: Asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensions
cites work: Nonlinear stability of shock fronts for a relaxation system in several space dimensions
cites work: \(L^ 1\) nonlinear stability of traveling waves for a hyperbolic system with relaxation
cites work: Q4892756
cites work: Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions
DOI: 10.1006/JDEQ.1998.3498
Recommended article: Relaxation limit for piecewise smooth solutions to systems of conservation laws
Recommended article: Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions
Recommended article: Q4240694
Recommended article: CONVERGENCE OF A RELAXATION APPROXIMATION TO A BOUNDARY VALUE PROBLEM FOR CONSERVATION LAWS
Recommended article: Nonlinear diffusive-dispersive limits for multidimensional conservation laws
Recommended article: Q4247550
Recommended article: Q4892756
Recommended article: Zero relaxation and dissipation limits for hyperbolic conservation laws
Recommended article: Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation
Recommended article: The strong relaxation limit of the multidimensional Euler equations
description: scientific article; zbMATH DE number 1271368