Stability of the selfsimilar dynamics of a vortex filament

TL;DR

This paper investigates the stability of selfsimilar solutions in vortex filament dynamics, demonstrating that small perturbations maintain their structure over time through detailed asymptotic analysis.

Contribution

It establishes the stability of selfsimilar vortex filament solutions under small perturbations, extending understanding of their long-term behavior.

Findings

Selfsimilar solutions are stable under small perturbations.

Precise asymptotics for tangent and normal vectors are derived.

Control of weighted norms for a related Schrödinger equation is achieved.

Abstract

In this paper we continue our investigation about selfsimilar solutions of the vortex filament equation, also known as the binormal flow (BF) or the localized induction equation (LIE). Our main result is the stability of the selfsimilar dynamics of small pertubations of a given selfsimilar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic 1-D Schr\"odinger equation, connected to the binormal flow by Hasimoto's transform.