# Erik Wahlén: Steady water waves with vorticity

**Time: **
Wed 2023-11-22 11.00 - 12.00

**Location: **
Albano, Cramérrummet

**Participating: **
Erik Wahlén (Lund)

**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.