Samuel Walsh: Desingularization of hollow vortices

Abstract:

A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid — think of it as a spinning bubble of air in water. In this talk, I will describe a general method for desingularizing non-degenerate translating, rotating, or stationary point vortex configurations into collections of steady hollow vortices. Through global bifurcation theory, moreover, these families can be extended to maximal curves of solutions that continue until the onset of a singularity. As specific applications, this machinery gives the first existence theory for co-rotating hollow vortex pairs and stationary hollow vortex tripoles, as well as a new construction of Pocklington’s classical co-translating hollow vortex pairs. All of these families extend into the non-perturbative regime, and we obtain a rather complete characterization of the limiting behavior along the global bifurcation curve.