Vortex Collapses for the Euler and Quasi-Geostrophic Models

The inviscid quasi-geostrophic equations, widely used to study the atmospheric dynamic, have many common points with the surface Euler equation. The main difference concerns the Biot and Savart law that involves a fractional laplace operator instead of a full laplace operator. From this observation, it is possible to extend the classical theory of point-vortices for the Euler equation to the quasi-geostrophic case. The point-vortex system is a system of differential hamiltonian first order equations that give account to the natural case where the vorticity is sharply concentrated around a finite number of points and then can be approximated by Dirac masses. Nevertheless, a point-vortex dynamic is well-defined as long as there are no collapses of vortices, due to the singularity of the vorticity kernels. This present talk aims at presenting some of the most recent results concerning the point-vortex systems both for the Euler and quasi-geostrophic models, with a focus on the vortex collapses.