Existence and stability of multidimensional shock fronts in the vanishing viscosity limit
type: Item
instance of: scholarly article
title: Existence and stability of multidimensional shock fronts in the vanishing viscosity limit
zbMATH Open document ID: 1072.35122
author: Olivier Guès
author: Mark Williams
author: Kevin Zumbrun
author: Guy Métivier
published in: Archive for Rational Mechanics and Analysis
publication date: 2005-04-14T00:00:00Z
review text: This interesting paper is devoted to the qualitative theory of shock waves resulting in \(N\times N\)-hyperbolic systems of partial differential equations describing conservation laws, i.e. \[ \partial_t u+\sum\limits_{j=1}^{d}\partial_jf_j(u)=0. \] Next, consider a parabolic viscous perturbation of that system \[ \partial_t u+\sum\limits_{j=1}^{d}\partial_jf_j(u)-\varepsilon \sum\limits_{j,k=1}^{d}\partial_j(B_{j,k}(u)\partial_ku)=0, \] where \(\varepsilon >0\) is a small number describing the rate of viscosity. The authors present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation. They obtain optimal estimates and an extension to the viscous case of the celebrated theorem of Majda on existence and stability of multidimensional shock waves. The main new idea is the derivation of maximal and optimal estimates for the linearization of the parabolic problem about a highly singular approximate solution. The key to the new approach is to use the full linearization of the parabolic problem. A transmission problem is obtained. It is shown that the linearized problem can be desingularized. Then optimal estimates are obtained by adding an appropriate extra boundary condition involving the front. This condition determines a local evolution rule for the viscous front. For the convenience of the reader, the authors collect the results about para-differential calculus which are used to prove the linear stability estimates.
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35L60
zbMATH DE Number: 2155991
zbMATH Keywords: small viscosity perturbation
zbMATH Keywords: maximal estimates
zbMATH Keywords: optimal estimates
zbMATH Keywords: para-differential calculus
MaRDI profile type: Publication
full work available at URL: s00205-004-0342-5
OpenAlex ID: W2018546573
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DOI: 10.1007/S00205-004-0342-5
description: scientific article; zbMATH DE number 2155991