Global bifurcation for corotating and counter-rotating vortex pairs

TL;DR

This paper extends the local bifurcation analysis of vortex pairs to a global setting, revealing that solutions either have zero angular velocity or self-intersect, using advanced topological and analytical methods.

Contribution

It constructs a global bifurcation curve for vortex pairs, overcoming singularities and topological challenges, which was not addressed in prior local analyses.

Findings

Global bifurcation curves are established for vortex pairs.

Solutions exhibit either zero angular velocity or self-intersection.

The analysis adapts the Buffoni-Toland global bifurcation theorem to singular problems.

Abstract

The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu in via a desingularization of a pair of point vortices. In this paper, we construct a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of a trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland, to allow for the singularity at the bifurcation point. For both the corotating and the counter-rotating pairs, along the global curve of solutions either the angular fluid velocity vanishes or the two patches self-intersect.