  We prove existence and asymptotic stability of the stationary solution for
the compressible Navier-Stokes equations for isentropic gas dynamics with a
density dependent diffusion in a bounded interval. We present the necessary
conditions to be imposed on the boundary data which ensure existence and
uniqueness of the steady state, and we subsequent investigate its stability
properties by means of the construction of a suitable Lyapunov functional for
the system. The Saint-Venant system, modeling the dynamics of a shallow
compressible fluid, fits into this general framework.
