Controllability for 2D water waves: effects of bottom topography and constant vorticity

Abstract

In this paper we consider two-dimensional water waves, under the action of gravity and surface tension. We prove a controllability result for irrotational waves in a fluid domain with finite depth and general bottom topography. The result holds for an open and dense set of bottom topographies in $H^{s+1/2}(\mathbb{T})$ (where $s$ is sufficiently large) that do not touch the free surface. We point out that the bottom topographies that we allow for are not necessarily small perturbations of the flat bottom case: this leads to many technical difficulties, since the eigenvalues of the Dirichlet-Neumann operator at a still free surface with general bottom topography are not explicit, nor are they necessarily close (for low frequencies) to the eigenvalues of the corresponding operator for the flat bottom case. In turn, this leads to a more involved argument to prove Ingham-type estimates, which are needed to prove observability, and motivates the restriction mentioned above on the admissible bottom topographies. We also prove a controllability result for waves with constant vorticity in a fluid domain with flat bottom topography.