  In this article, we consider the compressible Navier-Stokes equation with
density dependent viscosity coefficients and a term of capillarity introduced
by Coquel et al in \cite{5CR}. This model includes at the same time the
barotropic Navier-Stokes equations with variable viscosity coefficients,
shallow-water system and the model of Rohde. We first study the well-posedness
of the model in critical regularity spaces with respect to the scaling of the
associated equations. In a functional setting as close as possibleto the
physical energy spaces, we prove global existence of solutions close to a
stable equilibrium, and local in time existence for solutions with general
initial data. Uniqueness is also obtained.
