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Axi-symmetrization near point vortex solutions for the 2D Euler equation

Alexandru Ionescu, Hao Jia

TL;DR

This work establishes the nonlinear asymptotic stability of a two-dimensional point vortex for the Euler equations under small, Gevrey-smooth perturbations. The authors develop a renormalization framework using time-dependent, imbalanced weights and a carefully crafted change of variables to capture angular mixing and inviscid damping, despite the presence of a moving point vortex. They prove global regularity and show that the perturbed vorticity converges weakly to a radial Gevrey-2 profile centered at an asymptotic vortex position P_infty, with P(t) stabilizing rapidly. The analysis combines renormalization, elliptic estimates in variable coefficients, and a bootstrap argument to control high-frequency interactions and the coupling between the vortex motion and the perturbation. The resulting techniques offer a robust approach to nonlinear inviscid damping around vortex structures and may illuminate stability mechanisms in related 2D flow configurations.

Abstract

We prove a definitive theorem on the asymptotic stability of point vortex solutions to the full Euler equation in 2 dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as $t\to\infty$ to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping.

Axi-symmetrization near point vortex solutions for the 2D Euler equation

TL;DR

This work establishes the nonlinear asymptotic stability of a two-dimensional point vortex for the Euler equations under small, Gevrey-smooth perturbations. The authors develop a renormalization framework using time-dependent, imbalanced weights and a carefully crafted change of variables to capture angular mixing and inviscid damping, despite the presence of a moving point vortex. They prove global regularity and show that the perturbed vorticity converges weakly to a radial Gevrey-2 profile centered at an asymptotic vortex position P_infty, with P(t) stabilizing rapidly. The analysis combines renormalization, elliptic estimates in variable coefficients, and a bootstrap argument to control high-frequency interactions and the coupling between the vortex motion and the perturbation. The resulting techniques offer a robust approach to nonlinear inviscid damping around vortex structures and may illuminate stability mechanisms in related 2D flow configurations.

Abstract

We prove a definitive theorem on the asymptotic stability of point vortex solutions to the full Euler equation in 2 dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping.

Paper Structure

This paper contains 33 sections, 34 theorems, 410 equations.

Table of Contents

  1. Introduction
  2. Nonlinear asymptotic stability
  3. The general inviscid damping problem
  4. Adapted polar coordinates and precise results
  5. Main ideas of the proof
  6. Renormalization and time-dependent weights
  7. Variable coefficients
  8. Interaction with the point vortex
  9. Organization
  10. Renormalization, energy functionals, and the main proposition
  11. The main change of variables
  12. Equations for the coordinate functions
  13. Weights, energy functionals, and the bootstrap proposition
  14. The main bootstrap proposition
  15. Proofs of the main theorems
  16. ...and 18 more sections

Key Result

Theorem 1.1

Assume that $\kappa\in\mathbb{R}\backslash\{0\}, \lambda\in(0,\infty), M\in(1,\infty)$, and $\omega_0\in C^{\infty}_0(\mathbb{R}^2)$ satisfies the support property ${\rm supp}\,\omega_0\subseteq\{x\in\mathbb{R}^2:\,|x|\in[1/M,M]\}$. Assume that for a sufficiently small constant $\epsilon\leq\epsilon(\kappa,M,\lambda)$. Then there is a unique smooth global solution $(\omega,P)$ of the system Euler

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • ...and 47 more