  In this paper we study the motion of a surface gravity wave with viscosity.
In particular we prove two well-posedness results. On the one hand, we
establish the local solvability in Sobolev spaces for arbitrary dissipation. On
the other hand, we establish the global well-posedness in Wiener spaces for a
sufficiently large viscosity. These results are the first rigorous proofs of
well-posedness for the Dias, Dyachenko \& Zakharov system ({\em Physics Letters
A} 2008) modelling gravity waves with viscosity when surface tension is not
taken into account.
