  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.
