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

Abstract

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)=μϱ and λ(ϱ)=0 and a pressure law under the form p(ϱ)=aϱγ 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.