On Steady Water Waves and Flows with Vorticity in Three Dimensions

In this thesis we study the steady Euler equations in three dimensions where the solution is assumed to have nonvanishing vorticity. The thesis is based on three research papers. In the first and the third we study the steady Euler equations in the context of the water wave problem, which means we are solving a free boundary problem, while in the second paper we study the equations in a fixed cylinder-like domain. In the first paper we prove the existence of small amplitude doubly periodic waves when the velocity of the water is assumed to be a Beltrami field. Divergence free Beltrami fields are special solutions to the steady Euler equations where the velocity and vorticity are parallel. In the second paper we prove the existence of solutions of the steady Euler equations in cylinder-like domains, where the fluid flows through the domain, like water through a pipe. Here the vorticity is specified by two boundary conditions on the part of the surface where the fluid flows into the domain. In the third paper we also prove the existence of small amplitude doubly periodic waves, but with a different assumption on the vorticity. This assumption is more technical in nature and comes from magnetohydrodynamics. This theory is applicable because the governing equations for the magnetic field in a magnetohydrostatic equilibrium is equivalent to the steady Euler equations.