Erik Wahlén: Steady water waves with vorticity

Abstract:

The steady water wave problem is a classical topic in fluid mechanics
which has been studied for over two centuries. It concerns the steady
flow of an ideal fluid subject to gravity, bounded above by a free
surface. Mathematically it boils down to an elliptic free boundary
problem for the stream function. The most well-known situation is that
of irrotational flow, where the PDE is simply Laplace’s equation with
both Dirichlet and Neumann conditions at the free surface. In that case,
Stokes conjectured the existence of a highest wave with a peaked crest,
which was verified a century later in a seminal work by Amick, Fraenkel
and Toland. If one includes vorticity, Laplace’s equation changes to a
semilinear elliptic equation. This has some dramatic effects which are
not yet completely understood. In particular, it opens the door to
overhanging waves and `cat’s eye’ vortices. In my talk, I will report on
recent progress on these phenomena, including a new formulation of the
problem which allows for overhanging waves and has a structure which is
suitable for global bifurcation theory.

This is based on joint work with Jörg Weber.