Evolution by the vortex filament equation of curves with a corner

[Évolution par l’équation du tourbillon filamentaire de courbes à un coin]

Banica, Valeria

Journées équations aux dérivées partielles, (2013), p. 1-18 / Harvested from Numdam

Résumé

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 $\mathbb{R}^3$ 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.

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 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.

Détail

Publié le : 2013-01-01
DOI : https://doi.org/10.5802/jedp.97
Classification:  76B47,  35Q35,  35Q55,  35B35,  35P25

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