H\"older regularity for collapses of point vortices

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.