Viscosity effect on the degenerate lake equations

This paper concerns the effect of viscosity on the degenerate lake equations (anelastic limit) when the bottom topography vanishes on the shore. We establish the existence and uniqueness of a global weak solutions for various choices of viscosity term in weighted Sobolev spaces where the weight is assumed to be a power type weight. Our solution is constructed by adapting carefully the known works on the incompressible Navier-Stokes equations in unweighted context. Finally we study, in one case, the vanishing viscosity limit when the solution of the inviscid lake equations is regular enough generalizing the results by Martino et al. (2001) to the degenerate case.