Euler evolution of a concentrated vortex in planar bounded domains

TL;DR

This paper studies how a small concentrated vortex in a bounded domain evolves over time, showing it remains localized and behaves like a point vortex governed by the Kirchhoff-Routh equation.

Contribution

It proves that a small initial vortex remains localized with a diameter shrinking at a specific rate and converges to a point vortex following the Kirchhoff-Routh dynamics.

Findings

Vorticity support remains small over time.

Vorticity center converges to a point vortex.

Vortex diameter scales as ^{ extstylerac{1}{3}}.

Abstract

In this paper, we consider the time evolution of an ideal fluid in a planar bounded domain. We prove that if the initial vorticity is supported in a sufficiently small region with diameter $\varepsilon$, then the time evolved vorticity is also supported in a small region with diameter $d$, $d\leq C\varepsilon^{\alpha}$ for any $\alpha<\frac{1}{3}$, and the center of the vorticity tends to the point vortex, the motion of which is described by the Kirchhoff-Routh equation.