Boundary regularity of rotating vortex patches

TL;DR

This paper proves that the boundary of rotating vortex patches is infinitely smooth if they are close to a bifurcation circle, and convexity is guaranteed under C^2 proximity, using conformal mapping and singular integral estimates.

Contribution

It establishes boundary regularity and convexity conditions for rotating vortex patches near bifurcation points, extending previous understanding with new analytical techniques.

Findings

Boundary of vortex patches is C^infinity near bifurcation circle.

Convexity of patches is ensured close to bifurcation in C^2 norm.

Uses conformal mapping and Calderón-Zygmund operator estimates.

Abstract

We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class C^infinity provided the patch is close enough to the bifurcation circle in the Lipschitz norm. The rotating patch is convex if it is close enough to the bifurcation circle in the C^2 norm. Our proof is based on Burbea's approach to V-states. Thus conformal mapping plays a relevant role as well as estimating, on H\"older spaces, certain non-convolution singular integral operators of Calder\'on-Zygmund type.