Evolution by the vortex filament equation of curves with a corner

Banica, Valeria

Journées équations aux dérivées partielles (2013), article no. 1, 18 p.

Dans cet article de comptes rendus on présente une série de résultats sur la stabilité des solutions auto-similaires de l’équation du tourbillon filamentaire. Cette équation décrit un flot de courbes de ${ℝ}^3$ et est utilisée comme modèle pour l’évolution d’un tourbillon filamentaire dans un fluide. Le théorème principal donne, sous des hypothèses appropriées, l’existence et la description des solution engendrées par des courbes à un coin, sur temps positifs et négatifs. Le théorème compagnon décrit l’évolution des perturbations des solutions auto-similaires jusque’à formation d’une singularité en temps fini, et au-delà de ce temps. On va donner une esquisse des preuves. Ces résultats on été obtenus en collaboration avec Luis Vega.

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 ${ℝ}^3$ and it is used as a model for the evolution of a vortex filament in fluid mechanics. The main theorem give, 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.

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