  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.
