Rigidity of three-dimensional internal waves with constant vorticity

This paper studies the structural implications of constant vorticity for steady three-dimensional internal water waves in a channel. It is known that in many physical regimes, water waves beneath vacuum that have constant vorticity are necessarily two dimensional. The situation is more subtle for internal waves traveling along the interface between two immiscible fluids. When the layers have the same density, there is a large class of explicit steady waves with constant vorticity that are three-dimensional in that the velocity field is pointing in one horizontal direction while the interface is an arbitrary function of the other horizontal variable. We prove the following rigidity result: every three-dimensional traveling internal wave with bounded velocity for which the vorticities in the upper and lower layers are nonzero, constant, and parallel must belong to this family. If the densities in each layer are distinct, then in fact the flow is fully two dimensional. The proof is accomplished using an entirely novel but largely elementary argument that draws connection to the problem of uniquely reconstructing a two-dimensional velocity field from the pressure.