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

Recently, A. Vasseur and C. Yu have proved the existence of global entropy-weak solutions to the compressible Navier-Stokes equations with viscosities $\nu(\varrho)=\mu\varrho$ and $\lambda(\varrho)=0$ and a pressure law under the form $p(\varrho)=a\varrho^\gamma$ with $a>0$ and $\gamma>1$ 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 initiated by E. Feiersl, B.J. Jin and A. Novotny. We present some mathematical results related to the weak-strong uniqueness, convergence to a dissipative solution of compressible or incompressible Euler equations. As a by-product, this mathematically justifies the convergence of solutions of a viscous shallow water system to solutions of the inviscid shall-water system.