Entropy conditions for scalar conservation laws with discontinuous flux revisited

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TitreEntropy conditions for scalar conservation laws with discontinuous flux revisited
Type de publicationJournal Article
Year of Publication2015
AuteursAndreianov B, Mitrovic D
JournalANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
Volume32
Pagination1307-1335
Date PublishedNOV-DEC
Type of ArticleArticle
ISSN0294-1449
Mots-clésCrossing condition, discontinuous flux, Entropy solution, Inhomogeneous scalar conservation law, vanishing viscosity approximation, well-posedness
Résumé

We propose new entropy admissibility conditions for multidimensional hyperbolic scalar conservation laws with discontinuous flux which generalize one-dimensional Karlsen Risebro Towers entropy conditions. These new conditions are designed, in particular, in order to characterize the limit of vanishing viscosity approximations. On the one hand, they comply quite naturally with a certain class of physical and numerical modeling assumptions; on the other hand, their mathematical assessment turns out to be intricate. The generalization we propose is not only with respect to the space dimension, but mainly in the sense that the ``crossing condition'' of Karlsen, Risebro, and Towers (2003) [31] is not mandatory for proving uniqueness with the new definition. We prove uniqueness of solutions and give tools to justify their existence via the vanishing viscosity method, for the multi-dimensional spatially inhomogeneous case with a finite number of Lipschitz regular hypersurfaces of discontinuity for the flux function. (C) 2014 Elsevier Masson SAS. All rights reserved.

DOI10.1016/j.anihpc.2014.08.002