The global regularity of vortex patches revisited

TL;DR

This paper revisits the mathematical analysis of vortex patches, proving that their boundary regularity persists over time under certain transport equations with specific velocity fields.

Contribution

It establishes the persistence of boundary regularity for vortex patches in a broad class of transport equations with convolution-based velocity fields.

Findings

Boundary regularity of vortex patches is preserved over time.

The velocity field is modeled as a convolution of vorticity with a specific kernel.

The results apply to a large class of transport equations in the plane.

Abstract

We prove persistence of the regularity of the boundary of vortex patches for a large class of transport equations in the plane. The velocity field is given by convolution of the vorticity with an odd kernel, homogeneous of degree $-1$ and of class $C^2$ off the origin.