Evolution by the vortex filament equation of curves with a corner

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.