Long time confinement of vorticity around a stable stationary point vortex in a bounded planar domain

TL;DR

This paper investigates how vorticity remains confined around a stationary point vortex in a bounded planar domain, extending previous results to more general domains under conformal mapping conditions.

Contribution

It generalizes the power law confinement of vorticity around a stationary point vortex from the unit disk to any simply-connected bounded domain using conformal mappings.

Findings

Confinement law holds in general bounded domains.

Explicit examples illustrate the theoretical results.

Conditions involve conformal mapping properties.

Abstract

In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simply-connected bounded domain. The domain and the stationary point vortex must satisfy a condition expressed in terms of the conformal mapping from the domain to the unit disk. Explicit examples are discussed at the end.