The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper.
These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the
motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible
systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded
spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously.
First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist
globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular
with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during
this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish
within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish.
Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to
the non-vacuum equilibrium state exponentially in time.
Communicated by P. Constantin