  We prove the uniqueness and stability of entropy shocks to the isentropic
Euler systems among all vanishing viscosity limits of solutions to associated
Navier-Stokes systems. To take into account the vanishing viscosity limit, we
show a contraction property for any large perturbations of viscous shocks to
the Navier-Stokes system. The contraction estimate does not depend on the
strength of the viscosity. This provides a good control on the inviscid limit
process. We prove that, for any initial value, there exists a vanishing
viscosity limit to solutions of the Navier-Stokes system. The convergence holds
in a weak topology. However, this limit satisfies some stability estimates
measured by the relative entropy with respect to an entropy shock. In
particular, our result provides the uniqueness of entropy shocks to the shallow
water equation in a class of inviscid limits of solutions to the viscous
shallow water equations.
