Desingularization and global continuation for hollow vortices

Abstract

A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; it can therefore be interpreted as a spinning bubble of air in water. This paper gives a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. Our machinery simultaneously treats the translating, rotating, and stationary regimes. Through global bifurcation theory, we further obtain maximal curves of solutions that continue until the onset of a singularity. As specific examples, we give 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.