On self-similar singularity formation for the binormal flow

TL;DR

This paper provides a concise proof of the stability of self-similar solutions in the binormal flow, a model for vortex filament dynamics, using the connection to the 1-D cubic Schrödinger equation.

Contribution

It offers a simplified proof of stability results for self-similar solutions of the binormal flow in more restrictive cases, leveraging the Hasimoto transformation.

Findings

Established stability of self-similar solutions under certain conditions

Linked binormal flow solutions to 1-D cubic Schrödinger equation results

Provided a more concise proof compared to previous work

Abstract

The aim of this article is to establish a concise proof for a stability result of self-similar solutions of the binormal flow, in some more restrictive cases than in [5]. This equation, also known as the Local Induction Approximation, is a standard model for vortex filament dynamics, and its self-similar solution describes the formation of a corner singularity on the filament. Our approach strongly uses the link that Hasimoto pointed out in 1972 between the solution of the binormal flow and the one of the 1-D cubic Schr\"odinger equation, as well as the existence results associated to the latter.