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Two-Point Vortex Confinement in a simply connected domain

Slim Ibrahim, Ruixun Qin, Shengyi Shen

TL;DR

The work reduces the two-point vortex dynamics in a general simply connected domain to a Hamiltonian system in center and relative coordinates, then leverages higher-order expansions of the Green/Robin functions and KAM theory to prove long-time confinement near a stable stationary point. For nondegenerate stability, almost every small initial configuration remains confined for all time, while at the stability threshold confinement decays to a power-law in the small parameter. The results extend confinement analysis beyond symmetric domains by exploiting conformal mappings to the disk and explicit Diophantine-type frequency conditions. The approach highlights the interplay of complex analysis, Hamiltonian perturbation theory, and dynamical systems techniques in fluid vortex dynamics.

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.

Two-Point Vortex Confinement in a simply connected domain

TL;DR

The work reduces the two-point vortex dynamics in a general simply connected domain to a Hamiltonian system in center and relative coordinates, then leverages higher-order expansions of the Green/Robin functions and KAM theory to prove long-time confinement near a stable stationary point. For nondegenerate stability, almost every small initial configuration remains confined for all time, while at the stability threshold confinement decays to a power-law in the small parameter. The results extend confinement analysis beyond symmetric domains by exploiting conformal mappings to the disk and explicit Diophantine-type frequency conditions. The approach highlights the interplay of complex analysis, Hamiltonian perturbation theory, and dynamical systems techniques in fluid vortex dynamics.

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.

Paper Structure

This paper contains 8 sections, 7 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of the main Theorems
  4. Proof of Theorem \ref{['main result']}
  5. Proof of Theorem \ref{['main result 2']}
  6. Examples
  7. Acknowledgement
  8. Appendix

Key Result

Theorem 1.1

Given a simply connected domain $\Omega\subsetneq\mathbb{R}^2$ with $0\in \Omega^\circ$ the stationary point. Let $\varphi:\Omega\to\mathbb D$ be biholomorphic such that $\varphi(0)=0$ and the stability condition is satisfied. Then there exists $0<\varepsilon_{0}\ll1$, and a constant $\mu>0$ such that for any $0<\varepsilon\le\varepsilon_{0}$, if the vorticity strengths are such that $a_1+a_2\neq

Figures (1)

  • Figure 2: $\varphi(z) = a\left(\tan\left(iz\right)+\tan\left(i\frac{z}{2}\right)\right)$, $0$ is the unstable stationary point for $\frac{1}{2}<a<\frac{1}{\sqrt{3}}$ and stable stationary point for $a>\frac{1}{\sqrt{3}}$.

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1: bootstrap
  • Proposition A.1
  • Theorem A.2