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Comptes Rendus Mathématique
Volume 342, n° 11
pages 837-842 (juin 2006)
Doi : 10.1016/j.crma.2006.04.008
Received : 30 Mars 2006 ; 
Dynamics of multiple degree Ginzburg-Landau vortices
Dynamique des tourbillons de vorticité de degré multiple pour lʼéquation de Ginzburg-Landau

Fabrice Bethuel a, b , Giandomenico Orlandi c , Didier Smets a
a Laboratoire Jacques-Louis Lions, université de Paris 6, 4, place Jussieu, BC 187, 75252 Paris cedex 05, France 
b Institut Universitaire de France 
c Dipartimento di Informatica, Università di Verona, Strada le Grazie, 37134 Verona, Italy 


For the two-dimensional complex parabolic Ginzburg-Landau equation we prove that, asymptotically, vortices evolve according to a simple ordinary differential equation, which is a gradient flow of the Kirchhoff-Onsager functional. This convergence holds except for a finite number of times, corresponding to vortex collisions and splittings, which we describe carefully. The only assumption is a natural energy bound on the initial data. To cite this article: F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

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Nous montrons, pour lʼéquation de Ginzburg-Landau parabolique complexe en dimension deux, quʼasymptotiquement les tourbillons se déplacent suivant un flot gradient pour la fonctionnelle de Kirchhoff-Onsager. Cette convergence a lieu en dehors dʼun nombre fini dʼinstants qui correspondent aux éclatements et aux collisions des tourbillons, que nous décrivons en détail. Notre unique hypothèse sur les données initiales est une borne dʼénergie. Pour citer cet article : F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

The full text of this article is available in PDF format.

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