From point vortices to vortex patches in self-similar expanding configurations

TL;DR

This paper demonstrates that in 2D incompressible Euler flows, vortex patches can approximate self-similarly expanding point vortex configurations over time, with controlled growth of patch size and separation.

Contribution

It establishes a rigorous link between point vortex dynamics and vortex patches in self-similar expanding scenarios, extending classical vortex theory.

Findings

Vortex patches evolve like point vortices over time.

Patch size grows at most as t^{1/4+ε}.

Inter-vortex distance grows as √t.

Abstract

The main result is that given a generic self-similarly expanding configuration of 3 point vortices that start sufficiently far out, we can instead take compactly supported vorticity functions, and the resulting solution to 2D incompressible Euler will evolve like a nearby point vortex configuration for all time, with the size of the patches growing at most as $t^{1/4+\epsilon}$ and the distance between them growing as $\sqrt{t}$.