In recent years there has been a lot of progress on two-dimensional steady water waves with vorticity and three-dimensional waves with zero vorticity (irrotational flow). The three-dimensional case with vorticity remains largely unexplored. Part of the reason is that the problem doesn’t generally reduce to an elliptic free boundary problem in that case. In my talk I will discuss recent progress on this topic. In particular I will discuss one situation where the problem is elliptic, namely the case of doubly periodic waves on Beltrami flows. Here we have recently found small-amplitude gravity-capillary waves. I will also discuss an ongoing work where we find symmetric gravity-capillary waves with small vorticity, by treating the problem as elliptic-hyperbolic and using ideas from MHD theory.

This is joint work with E. Lokharu, D. S. Seth and K. Varholm.