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Stability of the Shape for Circular Vortex Filaments under Non-Symmetric Perturbations

Masashi Aiki, Mitsuo Higaki

Abstract

We establish the nonlinear orbital stability of circular vortex filaments governed by the Localized Induction Equation (LIE) under non-symmetric perturbations, within the framework of [Tani-Nishiyama, 1997]. This result extends the first author's recent work [Aiki, 2025] by removing symmetry assumptions on perturbations. While the circular filaments are known to be Lyapunov unstable due to linear growth of the translational mode, we prove that their shape remains globally stable modulo spatial translations and rotations about the symmetry axis. The crucial ingredient is a geometric stability lemma derived from the conservation of vector fluid impulse, which constrains the low-frequency modulations that are not covered by the relative energy.

Stability of the Shape for Circular Vortex Filaments under Non-Symmetric Perturbations

Abstract

We establish the nonlinear orbital stability of circular vortex filaments governed by the Localized Induction Equation (LIE) under non-symmetric perturbations, within the framework of [Tani-Nishiyama, 1997]. This result extends the first author's recent work [Aiki, 2025] by removing symmetry assumptions on perturbations. While the circular filaments are known to be Lyapunov unstable due to linear growth of the translational mode, we prove that their shape remains globally stable modulo spatial translations and rotations about the symmetry axis. The crucial ingredient is a geometric stability lemma derived from the conservation of vector fluid impulse, which constrains the low-frequency modulations that are not covered by the relative energy.
Paper Structure (7 sections, 6 theorems, 86 equations)

This paper contains 7 sections, 6 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Proof of Theorem \ref{['thm.main']}
  4. Estimates for the High-Frequency $|n|\ge2$ Modes
  5. Analysis of the $\pm1$ Modes
  6. Analysis of the Zero Mode
  7. End of the Proof and a Concrete Example

Key Result

Lemma 2.1

For a solution $\bm{x}=\bm{x}^R+\bm{\varphi}$ of eq.LIE, the following conservation laws hold:

Theorems & Definitions (8)

  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6: Geometric Stability Lemma
  • Lemma 2.7