On the dynamics of Navier–Stokes equations for a shallow water model

In this paper, we study a free boundary problem of one-dimensional compressible Navier–Stokes equations with a density-dependent viscosity, which include, in particular, a shallow water model. Under some suitable assumptions on the initial data, we obtain the global existence, uniqueness and the large time behavior of weak solutions. In particular, it is shown that a stationary wave pattern connecting a gas to the vacuum continuously is asymptotically stable for small initial general perturbations