On low regularity well-posedness of the binormal flow

TL;DR

This paper investigates the low regularity solutions of the binormal flow, a model for vortex filament evolution, using the Hasimoto transform and randomization techniques to establish existence and scattering results, including a novel limit measure identification.

Contribution

It extends the analysis of binormal flow solutions to lower regularity settings and introduces a new scattering result with a limit measure for the associated nonlinear Schrödinger equation.

Findings

Established existence of low regularity solutions for binormal flow.

Proved a scattering result for a quasi-invariance measure of the 1D cubic NLS.

Identified a limit measure not previously accessible in this context.

Abstract

We focus on a class of solutions of the binormal flow, model of the evolution of vortex filaments, that generate several corner singularities in finite time. This phenomenon has been studied earlier in the regular case, which in this context is in terms of the summability of the angles of the corners generated. Our goal here is to investigate the lower regularity case, using further the Hasimoto approach that allows to use the 1D cubic nonlinear Schr\"odinger to study the binormal flow. We first obtain a deterministic result by proving an existence result for general binormal flow solutions at low regularity. Then we obtain improved results on the above class of solutions by a suitable randomization of the curvature and torsion of the vortex filament. To do so, we prove a scattering result for a quasi-invariance measure associated with a suitable 1D cubic nonlinear Schr\"odinger equation that we consider of independent interest. An interesting feature of this result is that we are able to identify a limit measure, which is usually not possible when working on quasi-invariant Gaussian measures for Hamiltonian PDEs on bounded domains.