BD entropy and Bernis–Friedman entropy

Didier Bresch; Mathieu Colin; Khawla Msheik; Pascal Noble; Xi Song

doi:10.1016/j.crma.2018.11.009

Logic

BD entropy and Bernis–Friedman entropy

Presented by: Pierre-Louis Lions

Didier Bresch ¹; Mathieu Colin ²; Khawla Msheik ¹; Pascal Noble ³; Xi Song ²

¹ LAMA UMR5127 CNRS, Université Grenoble Alpes, Université Savoie Mont-Blanc, 73376 Le Bourget-du-Lac, France
² Équipe INRIA CARDAMOM, IMB Équipes EDP, 351, cours de la Libération, 33405 Talence, France
³ IMT, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 9, France

Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 1-6.

Abstract
Résumé

In this note, we propose in the full generality a link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis–Friedman (called BF) dissipative entropy introduced to study the lubrication equations. Different dissipative entropies are obtained playing with the drag terms on the viscous shallow-water equations. It helps for instance to prove the global existence of nonnegative weak solutions to the lubrication equations starting from the global existence of nonnegative weak solutions to appropriate viscous shallow-water equations.

Dans cette note, on propose un lien général entre la BD entropie introduite par D. Bresch et B. Desjardins pour les équations de Saint-Venant visqueuses et l'entropie dissipative de Bernis–Friedman (notée BF) introduite pour étudier les équations de lubrification. Différentes entropies dissipatives sont obtenues suivant le choix des termes de traînée sur Saint-Venant visqueux. Ce lien entre ces deux outils mathématiques aide, par exemple, à prouver l'existence de solutions faibles positives pour les équations de lubrification en partant de l'existence de solutions faibles positives pour des équations de Saint-Venant visqueuses bien choisies.

Received: 2018-11-07
Accepted: 2018-11-16
Published online: 2018-11-27

DOI: 10.1016/j.crma.2018.11.009

Author's affiliations:

Didier Bresch ¹; Mathieu Colin ²; Khawla Msheik ¹; Pascal Noble ³; Xi Song ²

¹ LAMA UMR5127 CNRS, Université Grenoble Alpes, Université Savoie Mont-Blanc, 73376 Le Bourget-du-Lac, France
² Équipe INRIA CARDAMOM, IMB Équipes EDP, 351, cours de la Libération, 33405 Talence, France
³ IMT, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 9, France

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     author = {Didier Bresch and Mathieu Colin and Khawla Msheik and Pascal Noble and Xi Song},
     title = {BD entropy and {Bernis{\textendash}Friedman} entropy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1--6},
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     language = {en},
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Didier Bresch; Mathieu Colin; Khawla Msheik; Pascal Noble; Xi Song. BD entropy and Bernis–Friedman entropy. Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 1-6. doi : 10.1016/j.crma.2018.11.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.009/

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