Two-Point Vortex Confinement in a simply connected domain

TL;DR

This paper studies how two-point vortices in a planar domain remain confined over time, demonstrating indefinite persistence of concentration and analyzing the effects of stability conditions on confinement duration.

Contribution

It provides a detailed analysis of vortex confinement times and their dependence on stability conditions and domain size, extending understanding of vortex dynamics.

Findings

Vortices remain confined indefinitely under certain conditions.

Confinement time follows a power law near stability boundaries.

Stability conditions critically influence vortex confinement duration.

Abstract

This paper investigates the vortex confinement property of the two-point vortex system in a planar domain. We compute the time over which initial point vortices around a stable stationary point remain within a slightly larger ball. In particular, we show that this concentration persists indefinitely regardless of the vorticity strengths. In the borderline of the stability condition, we show that this time becomes a power law, if in addition, one relaxes the size of the stability ball.