Overhanging water waves: new challenges and perspectives

Water waves are ubiquitous in nature, and understanding them is not only important for tourists during their beach holidays. This project deals with structural properties of water waves, where a special focus is on overhanging waves: breaking waves that surfers, for example, really like. We shall consider in particular travelling waves, propagating with unchanged shape and speed in a given direction. While basically the mathematical formulation of the underlying physics has been known for centuries, still many physical phenomena remain not fully studied and understood. This project deals with three different physical scenarios and tries to shed light on some open problems. First, in two dimensions (that is, the wave looks virtually unchanged in one, the third, spatial direction) we study the interplay of vortex patches with waves that have exactly one crest and have an almost undisturbed surface far off the crest. Here, a vortex patch is a region within the water where the fluid is revolving; think of the patterns formed when milk is poured into coffee or water is flushed down a toilet. Second, we consider axisymmetric waves, such as those created on a jet of water running out of a pipe. Again, we are interested how vorticity influences such waves. Third, in the fully three-dimensional situation, where patterns on the water surface emerge in all directions, not many rigorous mathematical results are available so far, and the project also tries to make substantial progress in this direction.