  We consider the free boundary problem for two layers of immiscible, viscous,
incompressible fluid in a uniform gravitational field, lying above a general
rigid bottom in a three-dimensional horizontally periodic setting. We establish
the global well-posedness of the problem both with and without surface tension.
We prove that without surface tension the solution decays to the equilibrium
state at an almost exponential rate; with surface tension, we show that the
solution decays at an exponential rate. Our results include the case in which a
heavier fluid lies above a lighter one, provided that the surface tension at
the free internal interface is above a critical value, which we identify. This
means that sufficiently large surface tension stabilizes the Rayleigh-Taylor
instability in the nonlinear setting. As a part of our analysis, we establish
elliptic estimates for the two-phase stationary Stokes problem.
