The initial value problem for the binormal flow with rough data

TL;DR

This paper investigates the initial value problem for the binormal flow with initial curves having a corner, establishing existence, uniqueness, and stability of solutions near self-similar solutions for both positive and negative times.

Contribution

It extends previous work by proving existence and uniqueness of solutions with rough initial data featuring a corner, using compactness and refined asymptotic analysis.

Findings

Existence of unique solutions with corner singularities for positive and negative times.

Solutions can be viewed as perturbations of self-similar solutions.

The approach avoids weighted conditions used in prior studies.

Abstract

In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique regular solution exists for strictly positive and strictly negative times. Moreover, this solution satisfies a weak version of the equation for all times and can be seen as a perturbation of a suitably chosen self-similar solution. Conversely, we also prove that if at time t = 1 a small regular perturbation of a self-similar solution is taken as initial condition then there exists a unique solution that at time t = 0 is regular except at a point where it has a corner with the same angle as the one of the self-similar solution. This solution can be extended for negative times. The proof uses the full strength of the previous papers [9], [2], [3] and [4] on the study of small perturbations of self-similar solutions. A compactness argument is used to avoid the weighted conditions we needed in [4], as well as a more refined analysis of the asymptotic in time and in space of the tangent and normal vectors.