  This paper concerns the dynamics of two layers of compressible, barotropic,
viscous fluid lying atop one another. The lower fluid is bounded below by a
rigid bottom, and the upper fluid is bounded above by a trivial fluid of
constant pressure. This is a free boundary problem: the interfaces between the
fluids and above the upper fluid are free to move. The fluids are acted on by
gravity in the bulk, and at the free interfaces we consider both the case of
surface tension and the case of no surface forces. We establish a sharp
nonlinear global-in-time stability criterion and give the explicit decay rates
to the equilibrium. When the upper fluid is heavier than the lower fluid along
the equilibrium interface, we characterize the set of surface tension values in
which the equilibrium is nonlinearly stable. Remarkably, this set is non-empty,
i.e. sufficiently large surface tension can prevent the onset of the
Rayleigh-Taylor instability. When the lower fluid is heavier than the upper
fluid, we show that the equilibrium is stable for all non-negative surface
tensions and we establish the zero surface tension limit.
