Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface

Abstract

Under consideration here are time-dependent three-dimensional stratified water flows of finite depth with a free surface and an interface (playing the role of an internal wave and separating two layers of constant and different densities). The main outcome of the paper is that, under the assumption that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other, we prove that bounded solutions to the three-dimensional equations are essentially two-dimensional. More precisely, the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. In addition, one of the horizontal components of the velocity field is constant throughout the flow and the vorticity also points in the direction orthogonal to the wave propagation direction. The rigid lid case is analyzed from the perspective of relaxed assumptions concerning the vorticity vectors. We also consider the viscous water wave problem case with normal and tangential stress conditions at the free surface and find that under similar conditions on the vorticity vectors, the conclusion about the two-dimensionality remains in place. Along the way, we also provide an explicit solution for the time-dependent three-dimensional inviscid water wave problem with an interface.