Geodesic
Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications
Didier Bresch1 ; Pascal Noble2 ; Jean-Paul Vila2
1 Laboratoire de Mathématiques UMR5127 CNRS Bâtiment le Chablais Université de Savoie Mont-Blanc 73376 Le Bourget du lac
2 Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, INSA, F-31077 Toulouse, France
ESAIM. Proceedings, Tome 58 (2017), pp. 40-57.

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This paper provides the full proof of the results announced by the authors in [C. R. Acad. Sciences (2016)]. We introduce an original relative entropy for compressible Navier-Stokes equations with density dependent viscosities and discuss some possible applications such as inviscid limit or low Mach number limit. We first consider the case µ(ϱ) = µϱ and λ(ϱ) = 0 and a pressure law under the form p(ϱ) = aϱγ with γ > 1, which corresponds in particular to the formulation of the viscous shallow water equations. We present some mathematical results related to the weak-strong uniqueness, the convergence to a dissipative solution of compressible or incompressible Euler equations. Moreover, we show the convergence of the viscous shallow water equations to the inviscid shallow water equations in the vanishing viscosity limit and further prove convergence to the incompressible Euler system in the low Mach limit. This extends results with constant viscosities recently initiated by E. Feireisl, B.J. Jin and A. Novotny in [J. Math. Fluid Mech. (2012)].
DOI : 10.1051/proc/201758040

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

1 Laboratoire de Mathématiques UMR5127 CNRS Bâtiment le Chablais Université de Savoie Mont-Blanc 73376 Le Bourget du lac
2 Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, INSA, F-31077 Toulouse, France 
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     author = {Didier Bresch and Pascal Noble and Jean-Paul Vila},
     title = {Relative entropy for compressible {Navier-Stokes} equations with density dependent viscosities and various applications},
     journal = {ESAIM. Proceedings},
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     doi = {10.1051/proc/201758040},
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Didier Bresch; Pascal Noble; Jean-Paul Vila. Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications. ESAIM. Proceedings, Tome 58 (2017), pp. 40-57. doi : 10.1051/proc/201758040. http://geodesic.mathdoc.fr/articles/10.1051/proc/201758040/

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