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

Abstract

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)].