  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 derive and study a new
asymptotic model for the motion of unidirectional viscous water waves. In
particular, we establish the global well-posedness in Sobolev spaces.
Furthermore, we also establish the global well-posedness and decay of a fourth
order PDE modelling bidirectional water waves with viscosity moving in deep
water with or without surface tension effects.
