H\"older regularity for collapses of point vortices

TL;DR

This paper investigates the regularity of point-vortex trajectories during collapses in fluid models, establishing optimal H"older continuity exponents and extending results to bounded domains with boundary effects.

Contribution

It proves the optimal H"older regularity of vortex trajectories during collapse for various models and extends the analysis to bounded domains with boundary considerations.

Findings

Vortex trajectories have H"older regularity up to collapse with exponent 1/(lpha+1).

The H"older exponent is shown to be optimal via explicit examples.

In bounded domains, vortex convergence and boundary regularity are established.

Abstract

The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of $\alpha$ models. In these models the kernel of the Biot-Savart law is a power function of exponent $-\alpha$. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a H\"older regularity up to, and including, the time of collapse. The H\"older exponent obtained is $1/(\alpha+1)$ and this exponent is proved to be optimal for all $\alpha$ by exhibiting an example of a $3$-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given point-vortex has an accumulation point in the interior of the domain as $t\to T$, then it converges towards this point and displays the same H\"older continuity property. A partial result for point-vortices that collapse with the boundary is also established : we prove that their distance to the boundary is H\"older regular.