Desingularization and global continuation for hollow vortices

TL;DR

This paper develops a general method to transform point vortex configurations into hollow vortices, analyzing their global behavior and providing new existence results for specific vortex structures.

Contribution

It introduces a novel desingularization technique for steady vortex configurations and constructs the first existence theory for certain hollow vortex pairs and tripoles.

Findings

Established a desingularization method for hollow vortices.

Proved existence of co-rotating hollow vortex pairs.

Characterized the limiting behavior of vortex families along bifurcation curves.

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.