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Stability of Lamb dipoles for odd-symmetric and non-negative initial disturbances without the finite mass condition

Ken Abe, Kyudong Choi, In-Jee Jeong

TL;DR

The paper proves orbital stability of the Chaplygin–Lamb Lamb dipole for the 2D Euler equations without the finite-mass restriction, under odd symmetry in $x_2$ and nonnegativity in the upper half-plane. It introduces a sharp energy inequality in the half-plane and a variational framework that characterizes minimizers of a constrained energy functional, showing they are translations of the Lamb dipole. Using concentration-compactness and an Euler–Lagrange analysis, the authors establish existence, compactness, and uniqueness of minimizers, which yields orbital stability around the Lamb dipole up to translations. The results bridge rigorous variational methods with vortex-dipole dynamics, aligning mathematical stability with observed dipole formation in 2D turbulence and providing tools to analyze long-time behavior without mass constraints.

Abstract

In this paper, we consider the stability of the Lamb dipole solution of the two-dimensional Euler equations in $\mathbb{R}^{2}$ and question under which initial disturbance the Lamb dipole is stable, motivated by experimental work on the formation of a large vortex dipole in two-dimensional turbulence. We assume (O) odd symmetry for the $x_2$-variable and (N) non-negativity in the upper half plane for the initial disturbance of vorticity, and establish the stability theorem of the Lamb dipole without assuming (F) finite mass condition. The proof is based on a new variational characterization of the Lamb dipole using an improved energy inequality.

Stability of Lamb dipoles for odd-symmetric and non-negative initial disturbances without the finite mass condition

TL;DR

The paper proves orbital stability of the Chaplygin–Lamb Lamb dipole for the 2D Euler equations without the finite-mass restriction, under odd symmetry in and nonnegativity in the upper half-plane. It introduces a sharp energy inequality in the half-plane and a variational framework that characterizes minimizers of a constrained energy functional, showing they are translations of the Lamb dipole. Using concentration-compactness and an Euler–Lagrange analysis, the authors establish existence, compactness, and uniqueness of minimizers, which yields orbital stability around the Lamb dipole up to translations. The results bridge rigorous variational methods with vortex-dipole dynamics, aligning mathematical stability with observed dipole formation in 2D turbulence and providing tools to analyze long-time behavior without mass constraints.

Abstract

In this paper, we consider the stability of the Lamb dipole solution of the two-dimensional Euler equations in and question under which initial disturbance the Lamb dipole is stable, motivated by experimental work on the formation of a large vortex dipole in two-dimensional turbulence. We assume (O) odd symmetry for the -variable and (N) non-negativity in the upper half plane for the initial disturbance of vorticity, and establish the stability theorem of the Lamb dipole without assuming (F) finite mass condition. The proof is based on a new variational characterization of the Lamb dipole using an improved energy inequality.

Paper Structure

This paper contains 22 sections, 18 theorems, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Lamb dipoles
  3. The statement of the main result
  4. Research on dipole vortices
  5. Physical backgrounds
  6. Large vortex dynamics
  7. Numerical works
  8. Small-scale formations
  9. Non-uniquness
  10. The idea of the proof: the new energy inequality
  11. Acknowledgements
  12. The variational principle
  13. The energy inequality
  14. The stream function estimates
  15. Minimization principle
  16. ...and 7 more sections

Key Result

Theorem 1.3

Let $0<\lambda, W<\infty$ and $P=c_0^{2}\pi W/\lambda$. The Lamb dipole $\omega_{L}$ is orbitally stable in the sense that for $\varepsilon>0$, there exists $\delta>0$ such that for $\zeta_0\in L^{2} (\mathbb{R}^{2}_{+})$ satisfying $x_2\zeta_{0}\in L^{1}(\mathbb{R}^{2}_{+})$, $\zeta_{0}\geq 0$, there exists a global weak solution $\zeta(t)$ of eq: Euler satisfying

Figures (3)

  • Figure 1: Streamlines of symmetric and asymmetric dipoles. Positive vorticity in red and negative vorticity in blue.
  • Figure 2: The emergence of a dipole vortex in a stratified flow created by a pulsed horizontal injection. Each photograph presents a top view and a side view. From FV94. The figure has been rotated by $90^\circ$. Licensed under CC BY 4.0.
  • Figure 3: The sets $\{0\leq \omega\leq \delta\}$ and $\{\omega> \delta\}$ in Proposition \ref{['p: omega']}

Theorems & Definitions (37)

  • Definition 1.1: Lamb dipole
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 27 more