Existence theory for damped gravity waves in a closed rectangular basin

We study existence and uniqueness of solutions of the equations for the free surface motion of an incompressible, irrotational fluid in a rectangular basin subject to vertical oscillation. After adding artificial damping, which leaves the flow irrotational but correctly represents the physical rate of energy loss at high wave numbers, we prove global existence and uniqueness results in the appropriate Sobolev spaces, provided that the initial data and forcing amplitudes have sufficiently small norms. Convergence of spatially discretized (finite-dimensional) projections is also discussed.