Global bifurcation of rotating vortex patches

TL;DR

This paper constructs continuous families of rotating vortex patches in 2D Euler equations, revealing structures like 'Cat's eyes' and potential corner formation, and proves boundary analyticity under regularity.

Contribution

It rigorously establishes the existence of large solution curves of rotating vortex patches and links flow structures to boundary regularity and potential singularities.

Findings

Existence of continuous solution curves with arbitrarily small angular velocity minima.

Presence of 'Cat's eyes'-type flow structures near the disk.

Analyticity of the boundary for sufficiently regular vortex patches.

Abstract

We rigorously construct continuous curves of rotating vortex patch solutions to the two-dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [WOZ84,Ove86] of singular limiting patches with 90$^\circ$ corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are "Cat's eyes"-type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular.