Vortex dynamics for 2D Euler flows with unbounded vorticity

Abstract

It is well-known that the dynamics of vortices in an ideal incompressible two-dimensional fluid contained in a bounded not necessarily simply connected smooth domain is described by the Kirchhoff--Routh point vortex system. In this paper, we revisit the classical problem of how well solutions to the Euler equations approximate these vortex dynamics and extend previous rigorous results to the case where the vorticity field is unbounded. More precisely, we establish estimates for the $2$-Wasserstein distance between the vorticity and the empirical measure associated with the point vortex dynamics. In particular, we derive an estimate on the order of weak convergence of the Euler solutions to the solutions of the point vortex system.