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

TL;DR

This paper proves that in bounded planar domains with smooth boundaries, almost all initial vortex configurations lead to globally existing solutions without collisions, extending previous results from the unit disk to more general domains.

Contribution

It extends global existence results for point vortex systems from the unit disk to general bounded domains with smooth boundaries, using new inequalities for Green's functions.

Findings

Almost every initial condition leads to global solutions.

No collisions occur between vortices or with the boundary.

Established inequalities for Green's functions in various domains.

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.