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,α 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 pointvortices 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.

Hölder regularity for collapses of point vortices

“…In the case of the Euler equations in a bounded simply connected smooth connected domain of R 2 with impermeability condition at the boundary, it is possible to proceed to an analogous construction [7] and get…”

“…Proof. A direct computation using the evolution equations for the point-vortex problem in bounded domains under developed form (7) gives…”

“…In [21], they prove that when the domain is a disc, the collapses are improbable in the sense that the set of initial datum leading to a collapse in finite time has a vanishing Lebesgue measure. This result has been extended to any bounded smooth domains (simply connected and multi-connected domains) by [7]. In [22, chap.4], they obtained the improbability of collapses when the domain is the whole plane, under the non-neutral hypothesis (presented hereafter).…”