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

TL;DR

This paper proves the nonlinear orbital stability of circular vortex filaments under non-symmetric perturbations, showing their shape remains stable despite linear instability, by leveraging geometric stability and fluid impulse conservation.

Contribution

It extends previous stability results by removing symmetry assumptions, establishing global shape stability of vortex filaments under general perturbations.

Findings

Circular vortex filaments are orbitally stable modulo translations and rotations.

The stability proof relies on a geometric lemma from fluid impulse conservation.

A connection between fluid impulse and isoperimetric inequality is established.

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. Finally, we relate the fluid impulse to an isoperimetric inequality, yielding a geometric constraint for closed filaments.