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 R3 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 in finite time, and beyond this time. We shall give a sketch of the proof. These results were obtained in collaboration with Luis Vega. In the first section we shall present the vortex filament equation. Then in the second section we shall give the topics we are interested in, and state our results. The sketch of the proofs will be given in the last section. 1. The Vortex Filament Equation The vortex filament equation (VFE) is the geometric flow of curves χ(t) in R3 governed by (1) χt = χx ∧ χxx. Here x stands for the arclength parameter of the curve χ(t). If one uses the Frenets system that characterize the tangent, normal and binormal vectors of a R3 curve in terms of the curvature and torsion, (2