Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-09T06:47:11.288Z Has data issue: false hasContentIssue false

Why many theories of shock waves are necessary: kinetic relations for non-conservative systems

Published online by Cambridge University Press:  30 January 2012

Christophe Berthon
Affiliation:
Laboratoire de Mathématiques Jean Leray, Centre National de la Recherche Scientifique, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes, France ([email protected])
Frédéric Coquel
Affiliation:
Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 75252 Paris, France ([email protected]; [email protected])
Philippe G. LeFloch
Affiliation:
Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 75252 Paris, France ([email protected]; [email protected])

Abstract

For a class of non-conservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial-value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to non-conservative systems of a similar concept introduced by Abeyaratne, Knowles and Truskinovsky for subsonic phase transitions and by LeFloch for non-classical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for non-conservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase-plane analysis of travelling-wave solutions associated with an augmented version of the non-conservative system. We illustrate with several examples that non-conservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of travelling waves which yields the corresponding kinetic function.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)