Weakly stable multidimensional shocks
type: Item
instance of: scholarly article
title: Weakly stable multidimensional shocks
zbMATH Open document ID: 1072.35120
author: Jean-François Coulombel
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2004-11-23T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78624
review text: The author considers the system of \(N\) conservation laws in \(\mathbb R \times \mathbb R ^d\) (time-space) \[ \sum\limits_{j=0}^{d}\partial_jf_j(u)=0, \] where \(x_0\equiv t\) is the time variable, \(x\equiv (x_1,x_2,\dots,x_d)\) is the space variable and \(\partial_j\equiv \partial /\partial x_j\). The fluxes \(f_0,f_1,\dots,f_d\) are \(C^{\infty }\) functions defined on an open set \(U\subset \mathbb R ^N\) with values in \(\mathbb R ^N\). The author studies the linear stability of multidimensional shock waves for the system under consideration in the case where Majda's uniform stability condition is violated. An energy estimate is obtained. It is shown that the solutions to the linearized problem have singularities localized along bicharacteristic curves originating from the boundary. The application to isentropic gas dynamics is discussed.
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35L50
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 35L35
Mathematics Subject Classification ID: 76N15
zbMATH DE Number: 2117889
zbMATH Keywords: symmetrizers
zbMATH Keywords: linear stability
zbMATH Keywords: isentropic gas dynamics
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
OpenAlex ID: W2025684803
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DOI: 10.1016/J.ANIHPC.2003.04.001
description: scientific article; zbMATH DE number 2117889