Evolution by the vortex filament equation of curves with a corner

TL;DR

This paper surveys results on the stability and evolution of curves with corners under the vortex filament equation, a geometric flow modeling vortex filaments in fluid mechanics, including existence, description, and perturbation analysis of solutions.

Contribution

It presents new theorems on the existence and evolution of solutions starting from curves with corners, extending understanding of singularity formation in vortex filament dynamics.

Findings

Existence of solutions from curves with corners for positive and negative times.

Description of the evolution of perturbations of self-similar solutions.

Analysis of singularity formation and solution behavior at infinite time.

Abstract

In this proceedings article we shall survey a series of results on the stability of self-similar solutions of the vortex filament equation. This equation is a geometric flow for curves in $\mathbb R^3$ and it is used as a model for the evolution of a vortex filament in fluid mechanics. The main theorem gives, under suitable assumptions, the existence and description of solutions generated by curves with a corner, for positive and negative times. Its companion theorem describes the evolution of perturbations of self-similar solutions up to a singularity formation infinite time, and beyond this time. We shall give a sketch of the proof. These results were obtained in collaboration with Luis Vega.