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. Finally, we relate the fluid impulse to an isoperimetric inequality, yielding a geometric constraint for closed filaments.
Keywords
Cite
@article{arxiv.2602.18155,
title = {Stability of the Shape for Circular Vortex Filaments under Non-Symmetric Perturbations},
author = {Masashi Aiki and Mitsuo Higaki},
journal= {arXiv preprint arXiv:2602.18155},
year = {2026}
}
Comments
Improved the estimate in Theorem 1.1 along with a simplified proof. Added an appendix. Expanded the abstract and introduction accordingly, and refined overall wording. 13 pages