Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications
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)].
Affiliations des auteurs :
Didier Bresch 1 ; Pascal Noble 2 ; Jean-Paul Vila 2
@article{EP_2017_58_a3,
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},
pages = {40--57},
year = {2017},
volume = {58},
doi = {10.1051/proc/201758040},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201758040/}
}
TY - JOUR AU - Didier Bresch AU - Pascal Noble AU - Jean-Paul Vila TI - Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications JO - ESAIM. Proceedings PY - 2017 SP - 40 EP - 57 VL - 58 UR - http://geodesic.mathdoc.fr/articles/10.1051/proc/201758040/ DO - 10.1051/proc/201758040 LA - en ID - EP_2017_58_a3 ER -
%0 Journal Article %A Didier Bresch %A Pascal Noble %A Jean-Paul Vila %T Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications %J ESAIM. Proceedings %D 2017 %P 40-57 %V 58 %U http://geodesic.mathdoc.fr/articles/10.1051/proc/201758040/ %R 10.1051/proc/201758040 %G en %F EP_2017_58_a3
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
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