Symmetric doubly periodic gravity-capillary waves with small vorticity

TL;DR

This paper constructs small amplitude, symmetric, doubly periodic gravity-capillary water waves with small vorticity in three dimensions, addressing challenges posed by the free boundary and non-elliptic nature of the problem.

Contribution

It introduces a novel bifurcation approach for free boundary water waves with vorticity, overcoming regularity loss and non-ellipticity issues.

Findings

Existence of symmetric doubly periodic gravity-capillary waves with small vorticity.

Application of a modified Crandall-Rabinowitz bifurcation method to free boundary problems.

Handling of non-elliptic free boundary conditions in water wave theory.

Abstract

We construct small amplitude gravity-capillary water waves with small nonzero vorticity, in three spatial dimensions, bifurcating from uniform flows. The waves are symmetric, and periodic in both horizontal coordinates. The proof is inspired by Lortz' construction of magnetohydrostatic equilibria in reflection-symmetric toroidal domains. It relies on a global representation of the vorticity as the cross product of two gradients, and on prescribing a functional relationship between the Bernoulli function and the orbital period of the water particles. The presence of the free surface introduces significant new challenges. In particular, the resulting free boundary problem is not elliptic, and the involved maps incur a loss of regularity under Fr\'echet differentiation. Nevertheless, we show that a version of the Crandall-Rabinowitz local bifurcation method applies, by carefully tracking the loss of regularity.