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Global dynamics of a single vortex ring

Dengjun Guo, In-Jee Jeong, Lifeng Zhao

Abstract

We study the global-in-time dynamics of vortex rings for the three-dimensional incompressible Euler equations, under the assumption of axisymmetric flows without swirl. For a broad class of initial data sharing only the macroscopic invariants with a thin vortex ring, we prove that the vorticity remains sharply concentrated and propagates along the symmetry axis with leading-order speed given by the Kelvin--Hicks formula, providing the first global-in-time validation of the vortex filament conjecture for a single vortex ring arising from generic initial data. We further identify a universal filamentation mechanism driven by the competition between rapid core translation and slower local induction. This mechanism gives linear-in-time stretching of the vortex support under very general assumptions on the data, yielding dynamical instability of any thin vortex ring configurations in the $W^{2,\infty}$ norm.

Global dynamics of a single vortex ring

Abstract

We study the global-in-time dynamics of vortex rings for the three-dimensional incompressible Euler equations, under the assumption of axisymmetric flows without swirl. For a broad class of initial data sharing only the macroscopic invariants with a thin vortex ring, we prove that the vorticity remains sharply concentrated and propagates along the symmetry axis with leading-order speed given by the Kelvin--Hicks formula, providing the first global-in-time validation of the vortex filament conjecture for a single vortex ring arising from generic initial data. We further identify a universal filamentation mechanism driven by the competition between rapid core translation and slower local induction. This mechanism gives linear-in-time stretching of the vortex support under very general assumptions on the data, yielding dynamical instability of any thin vortex ring configurations in the norm.
Paper Structure (22 sections, 14 theorems, 120 equations, 1 figure)

This paper contains 22 sections, 14 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Axisymmetric Euler equation
  3. Main Results
  4. Ideas of the proof
  5. Literature on vortex rings
  6. Construction of steady vortex rings
  7. Interaction of multiple vortex rings and point vortices
  8. Global dynamics of a single vortex ring with general data
  9. Filamentation and infinite time norm growth
  10. Contribution of the present work
  11. Notation
  12. Organization of the paper
  13. Concentration
  14. Preliminaries
  15. Concentration and confinement in $r$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Given $r_{0}, \mu, c_{1}, c_{2}, c_{3}, c_{4} > 0$, there exist $\epsilon_0, C_{0} > 0$ depending on these parameters such that the following holds. For any $0< \epsilon \le \epsilon_0$, assume that $w_{0,\epsilon} \in L^{\infty}(\mathbb H)$ and satisfies: Let $w_{\epsilon}(\cdot,t)$ be the corresponding solution to the axisymmetric Euler equation eq axeuler. Then for any $t \in \mathbb R$, $w_{\

Figures (1)

  • Figure 1: Illustration for an example of $w_{0,\epsilon}$ satisfying the assumptions of Theorem \ref{['thm filamentation']}. The black and gray regions represent the support of $w_{0,m,\epsilon}$ and $w_{0,d,\epsilon}$, where the vorticity has size $O(\epsilon^{-2})$ and $O(1)$, respectively.

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 20 more