Relative entropy for compressible Navier–Stokes equations with density-dependent viscosities and applications

Didier Bresch; Pascal Noble; Jean-Paul Vila

doi:10.1016/j.crma.2015.10.003

Partial differential equations

Relative entropy for compressible Navier–Stokes equations with density-dependent viscosities and applications

Presented by: Gilles Lebeau

Didier Bresch ¹; Pascal Noble ²; Jean-Paul Vila ²

¹ LAMA – UMR5127 CNRS, Bât. Le Chablais, Campus scientifique, 73376 Le Bourget-du-Lac, France
² IMT, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 9, France

Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 45-49.

Abstract
Résumé

Recently A. Vasseur and C. Yu have proved (see A. Vasseur, C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations, arXiv:1501.06803, 2015) the existence of global entropy-weak solutions to the compressible Navier–Stokes equations with viscosities $ν (ϱ) = μ ϱ$ and $λ (ϱ) = 0$ and a pressure law under the form $p (ϱ) = a ϱ^{γ}$ with $a > 0$ and $γ > 1$ constants. In this note, we propose a non-trivial relative entropy for such system in a periodic box and give some applications. This extends, in some sense, results with constant viscosities recently initiated by E. Feireisl, B.J. Jin and A. Novotny in [J. Math. Fluid Mech. (2012)]. We present some mathematical results related to the weak–strong uniqueness, the convergence to a dissipative solution to compressible or incompressible Euler equations. As a by-product, this mathematically justifies the convergence of solutions to a viscous shallow-water system to solutions to the inviscid shallow-water system.

Récemment, A. Vasseur et C. Yu ont prouvé (voir A. Vasseur, C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations, arXiv:1501.06803, 2015) l'existence globale de solutions faibles entropiques des équations de Navier–Stokes compressibles avec des viscosités $ν (ϱ) = μ ϱ$ , $λ (ϱ) = 0$ et une pression du type $p (ϱ) = a ϱ^{γ}$ , avec $a > 0$ et $γ > 1$ deux constantes. Dans cette note, on propose une entropie relative originale pour un tel système, avec cette dépendance des viscosités en la densité, et on donne quelques applications. Ceci étend les résultats avec viscosités constantes initiés par E. Feireisl, B.J. Jin and A. Novotny dans [J. Math. Fluid Mech. (2012)]. On présente quelques résultats liés à l'unicité faible–fort, la convergence vers une solution dissipative d'Euler compressible. Ceci justifie en particulier la convergence d'un système de Saint-venant avec viscosité vers son analogue non visqueux.

Received: 2015-05-21
Accepted: 2015-10-05
Published online: 2015-11-06

DOI: 10.1016/j.crma.2015.10.003

Author's affiliations:

Didier Bresch ¹; Pascal Noble ²; Jean-Paul Vila ²

¹ LAMA – UMR5127 CNRS, Bât. Le Chablais, Campus scientifique, 73376 Le Bourget-du-Lac, France
² IMT, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 9, France

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     author = {Didier Bresch and Pascal Noble and Jean-Paul Vila},
     title = {Relative entropy for compressible {Navier{\textendash}Stokes} equations with density-dependent viscosities and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {45--49},
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     volume = {354},
     number = {1},
     year = {2016},
     doi = {10.1016/j.crma.2015.10.003},
     language = {en},
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Didier Bresch; Pascal Noble; Jean-Paul Vila. Relative entropy for compressible Navier–Stokes equations with density-dependent viscosities and applications. Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 45-49. doi : 10.1016/j.crma.2015.10.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.003/

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