Mathematics

The steady water wave problem concerns travelling wave solutions to the incompressible Euler equations with a free boundary. The vorticity is the curl of the velocity field. In the irrotational case, that is when the vorticity vanishes, the problem reduces to an elliptic free boundary problem for the harmonic velocity potential. In this case, many results have been obtained in both two and three dimensions (modelling waves which are either uniform perpendicular to the wave motion or doubly periodic). In recent years there have also been a lot of interest in the two-dimensional problem with vorticity, which can be used to model interactions of surface waves with ocean currents. This problem can be formulated as a semilinear elliptic free boundary problem for the stream function. On the other hand, there are very few results on the corresponding three-dimensional problem. This is partly due to the loss of ellipticity. In my talk, I will give an overview of the two-dimensional problem and report on some recent progress on the three-dimensional case.