In this talk we prove the existence of small amplitude three-dimensional doubly periodic steady water waves with vorticity. The domain of the fluid has finite depth, and pattern of the waves is non-symmetric with respect to the propagation direction of the waves; moreover, the relative velocity field is a Beltrami field, meaning that the vorticity is proportional to the velocity. The main difficulty arises from the presence of vorticity, which in turn leads to a careful study of a generalized Dirichlet-Neumann operator; moreover, in the case of vanishing surface tension, we have to overcome small-divisors issues.
This is a joint work with M. Groves (Saarland University), D. Nilsson and E. Wahlén (Lund University).
