  The motion of the free surface of an incompressible fluid is a very active
research area. Most of these works examine the case of an inviscid fluid.
However, in several practical applications, there are instances where the
viscous damping needs to be considered. In this paper we study an asymptotic
model for a viscous free boundary flow. In particular, we establish the global
well-posedness in Sobolev spaces of a fourth order PDE modelling water waves
with viscosity moving in deep water with or without surface tension effects.
Furthermore, we also prove the decay of these solutions towards the equilibrium
state.
