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Recently, A. Vasseur and C. Yu have proved the existence of globalentropy-weak solutions to the compressible Navier-Stokes equations withviscosities $\nu(\varrho)=\mu\varrho$ and $\lambda(\varrho)=0$ and a pressurelaw 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 suchsystem in a periodic box and give some applications. This extends, in somesense, results with constant viscosities initiated by E. Feiersl, B.J. Jin andA. Novotny.  We present some mathematical results related to the weak-strong uniqueness,convergence to a dissipative solution of compressible or incompressible Eulerequations. As a by-product, this mathematically justifies the convergence ofsolutions of a viscous shallow water system to solutions of the inviscidshall-water system.

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