On classical solutions to 2D Shallow water equations with degenerate viscosities

In this paper, the $2$-D isentropic Navier-Stokes systems for compressible fluids with density-dependent viscosity coefficients are considered. In particular, we assume that the viscosity coefficients are proportional to density. These equations, including several models in $2$-D shallow water theory, are degenerate when vacuum appears. We introduce the notion of regular solutions and prove the local existence of solutions in this class allowing the initial vacuum in the far field. This solution is further shown to be stable with respect to initial data in $H^2$ sense. A Beal-Kato-Majda type blow-up criterion is also established.