Kinetic formulation for systems of two conservation laws and elastodynamics
type: Item
instance of: scholarly article
title: Kinetic formulation for systems of two conservation laws and elastodynamics
zbMATH Open document ID: 0980.35092
published in: Archive for Rational Mechanics and Analysis
publication date: 2001-05-15T00:00:00Z
review text: The authors introduce the so-called kinetic formulation for the system of elastodynamics \[ u_t - v_x = 0,\qquad v_t - \sigma(u)_x = 0 \] in one space variable, where \(\sigma\) is an increasing nonlinear function with exactly one inflection point located at zero. The kinetic formulation allows to characterize the entropy solutions of the problem in terms of the kinetic functions \(\Theta(u,v,\xi)\), \(\overline \Theta(u,v,\xi)\) satisfying the system \[ \partial_t \Theta + \partial_x [ (-a(u) \kappa + a(u) (1- \kappa)) \Theta ] = \partial_{\xi} m , \] \[ \partial_t \overline \Theta + \partial_x [ (-a(u) \overline \kappa + a(u) (1- \overline \kappa)) \overline \Theta ] = \partial_{\xi} \overline m, \] where \(a(u) = (\sigma'(u))^{1\over 2}\), \(\kappa\), \(\overline \kappa\) are certain functions of \(u,v\) and of the kinetic variable \(\xi\), and \(m\) and \(\overline m\) are measures. This approach gives a full characterization of convex entropies related to the system in question and simplifies considerably the existing theory based on the concept of the Young measures. The paper provides also a very nice and self-contained exposition of both the kinetic approach to systems of conservation laws and the nonlinear wave equation theory.
reviewed by: Eduard Feireisl
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 74J30
Mathematics Subject Classification ID: 35Q72
zbMATH DE Number: 1556302
zbMATH Keywords: nonlinear elastodynamics
zbMATH Keywords: one space variable
zbMATH Keywords: entropy solution
author: Ben{ôı}t Perthame
author: Athanasios E. Tzavaras
MaRDI profile type: Publication
full work available at URL: s002050000109
OpenAlex ID: W2051014922
DOI: 10.1007/S002050000109
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description: scientific article; zbMATH DE number 1556302