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Long-time Confinement near Special Vortex Crystals

Martin Donati

TL;DR

This work proves long-time confinement of vorticity for 2D Euler flows when the initial vorticity is concentrated near a special vortex crystal comprising $N-1$ exterior vortices at the vertices of a regular $(N-1)$-gon and a central vortex of intensity $\\gamma_N$. By linking desingularized vortex blobs to the point-vortex system and imposing a strong linear stability condition together with a noncollapse constraint, the authors derive a quantitative confinement window: if the initial support radius is $\\varepsilon$, the support remains within $\\varepsilon^{\\beta}$ for a time at least $\\varepsilon^{-\\alpha}$ with $\\beta<\\tfrac12$ and $\\alpha<\\min(\\beta/2, 2-4\\beta)$. The central result asserts the existence of a unique $\\gamma_N$ for each $N\ge4$, given by $\\gamma_N=(N-2)(N-6)/12$, such that the configuration is linearly and nonlinearly stable, and thus enjoys long-time confinement; in the special pentagon case ($N=5$) the central vortex intensity vanishes. These findings extend prior confinement results beyond single-vortex or rapidly separating regimes by exploiting the strain properties of polygonal vortex crystals.

Abstract

In this paper, we control the growth of the support of particular solutions to the Euler two-dimensional equations, whose vorticity is concentrated near special vortex crystals. These vortex crystals belong to the classical family of regular polygons with a central vortex, where we choose a particular intensity for the central vortex to have strong stability properties. A special case is the regular pentagon with no central vortex which also satisfies the stability properties required for the long-time confinement to work.

Long-time Confinement near Special Vortex Crystals

TL;DR

This work proves long-time confinement of vorticity for 2D Euler flows when the initial vorticity is concentrated near a special vortex crystal comprising exterior vortices at the vertices of a regular -gon and a central vortex of intensity . By linking desingularized vortex blobs to the point-vortex system and imposing a strong linear stability condition together with a noncollapse constraint, the authors derive a quantitative confinement window: if the initial support radius is , the support remains within for a time at least with and . The central result asserts the existence of a unique for each , given by , such that the configuration is linearly and nonlinearly stable, and thus enjoys long-time confinement; in the special pentagon case () the central vortex intensity vanishes. These findings extend prior confinement results beyond single-vortex or rapidly separating regimes by exploiting the strain properties of polygonal vortex crystals.

Abstract

In this paper, we control the growth of the support of particular solutions to the Euler two-dimensional equations, whose vorticity is concentrated near special vortex crystals. These vortex crystals belong to the classical family of regular polygons with a central vortex, where we choose a particular intensity for the central vortex to have strong stability properties. A special case is the regular pentagon with no central vortex which also satisfies the stability properties required for the long-time confinement to work.

Paper Structure

This paper contains 18 sections, 15 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Modelling and main result
  3. Detailed hypotheses and results
  4. Hypotheses on the configuration
  5. The confinement result
  6. Existence of the configuration
  7. Stability of relative equilibria
  8. Confinement
  9. Plan of the proof
  10. Preliminary tools
  11. Proof of Lemma \ref{['lem:lip_amélioré']}
  12. Proof of Lemma \ref{['lem:moments']}
  13. Proof of Lemma \ref{['lem:4n']}
  14. Choosing the intensity of the central vortex
  15. Reduction by symmetries
  16. ...and 3 more sections

Key Result

Theorem 2.2

Let $(\omega_0^\varepsilon)_{\varepsilon>0}$ satisfying Hypothesis hyp:omega for some $z_1,\ldots,z_N$ and $\gamma_1,\ldots,\gamma_N$ giving rise to a global solution of the point-vortex dynamics PVS such that Then for any $\beta<1/2$ there exists a constant $C$ such that for any $\varepsilon > 0$ small enough, the solution $\omega^\varepsilon$ of the problem eq:du_pb satisfies $\tau_{\varepsilon

Figures (1)

  • Figure 1: Various polygonal vortex crystals of radius 1 and exterior intensities 1, where $\gamma_N$ designates the intensity of the central vortex.

Theorems & Definitions (20)

  • Theorem 2.2: marchiorobutta2018
  • Theorem 2.3: marchiorobutta2018
  • Theorem 2.3: marchiorobutta2018
  • Definition 2.4
  • Theorem 2.5: Cabral_Schmidt_1999_Stability_N+1_vortex
  • Theorem 2.6
  • Remark 2.7
  • Proposition 3.2: Roberts_2013_Stability_of_Relative_Eq
  • Theorem 3.4
  • Theorem 3.5
  • ...and 10 more