Hölder 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 α models. This consists in a Biot-Savart law with a kernel being a power function of exponent −α. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the vorticies have a regularity Hölder at T the time of collapse. The Hölder exponent obtained is 1/(α + 1) and this exponent is proved to be optimal for all α 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 vortex has an adherence point in the interior of the domain as t → T , then it converges towards this point and show the same Hölder continuity property.