The positive entropy production property for augmented nonlinear hyperbolic models

Philippe G. LeFloch; Allen M. Tesdall

doi:10.5802/crmath.278

Analyse numérique, Équations aux dérivées partielles

The positive entropy production property for augmented nonlinear hyperbolic models

Philippe G. LeFloch ¹ ; Allen M. Tesdall ²

¹ Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique Sorbonne Université, 4 Place Jussieu, 75252 Paris, France.
² Department of Mathematics, City University of New York, College of Staten Island, and Physics Program, The Graduate Center, City University of New York, New York, U.S.A.

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 35-46.

Résumé

Given a first-order nonlinear hyperbolic system of conservation laws endowed with a convex entropy-entropy flux pair, we consider the class of weak solutions containing shock waves depending upon some small scale parameters. In this Note, after introducing a notion of positive entropy production property that involves test-functions (rather than solutions), we define and derive several classes of entropy-dissipating augmented models, as we call them, which involve (possibly nonlinear) second- and third-order augmentation terms. Such terms typically arise in continuum physics and model viscosity and other high-order effects in a fluid. By introducing a new notion of positive entropy production that concerns general functions rather than solutions, we can easily check the entropy-dissipating property for a broad class of augmented models. The weak solutions associated with the corresponding zero-limit may contain (nonclassical undercompressive) shocks whose selection is determined from these high-order effects, for instance by using traveling wave solutions. Having a classification of the underlying models, as we propose, is essential for developing a general shock wave theory.

Reçu le : 2019-12-14
Révisé le : 2020-05-07
Accepté le : 2021-10-03
Publié le : 2022-01-26

DOI : 10.5802/crmath.278

Affiliations des auteurs :

Philippe G. LeFloch ¹ ; Allen M. Tesdall ²

¹ Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique Sorbonne Université, 4 Place Jussieu, 75252 Paris, France.
² Department of Mathematics, City University of New York, College of Staten Island, and Physics Program, The Graduate Center, City University of New York, New York, U.S.A.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The positive entropy production property for augmented nonlinear hyperbolic models},
     journal = {Comptes Rendus. Math\'ematique},
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     publisher = {Acad\'emie des sciences, Paris},
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     year = {2022},
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     language = {en},
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Philippe G. LeFloch; Allen M. Tesdall. The positive entropy production property for augmented nonlinear hyperbolic models. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 35-46. doi : 10.5802/crmath.278. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.278/

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