2D point vortex dynamics in bounded domains: global existence for almost every initial data

Abstract

In this paper, we prove that in bounded planar domains with $C^{2,\alpha}$ boundary, for almost every initial condition in the sense of the Lebesgue measure, the point vortex system has a global solution, meaning that there is no collision between two point-vortices or with the boundary. This extends the work previously done in [13] for the unit disk. The proof requires the construction of a regularized dynamics that approximates the real dynamics and some strong inequalities for the Green's function of the domain. In this paper, we make extensive use of the estimates given in [7]. We establish our relevant inequalities first in simply connected domains using conformal maps, then in multiply connected domains.