Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature

Fabrice Bethuel; Giandomenico Orlandi; Didier Smets

doi:10.1016/S1631-073X(03)00167-5

Partial Differential Equations

Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature
[Convergence de l'équation de Ginzburg–Landau parabolique vers un mouvement par courbure moyenne]

Présenté par : Haı̈m Brezis

Fabrice Bethuel ^1,² ; Giandomenico Orlandi ³ ; Didier Smets ¹

¹ Laboratoire Jacques-Louis Lions, Université de Paris 6, 4, place Jussieu, BC 187, 75252 Paris, France
² Institut universitaire de France
³ Dipartimento di Informatica, Università di Verona, Strada le Grazie, 37134 Verona, Italy

Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 719-723.

Résumé
Abstract

Nous présentons de nouveaux résultats concernant l'étude asymptotique du flot de la chaleur pour l'énergie de Ginzburg–Landau. En particulier, nous montrons que, lorsque le paramètre ε tend vers 0, la vorticité évolue selon un mouvement par courbure moyenne, dans un sens faible introduit par Brakke. Notre seule hypothèse concerne une borne naturelle portant sur l'énergie de la condition initiale. Dans certains cas, nous montrons également la convergence vers un mouvement par courbure moyenne dans un sens plus fort dû à Ilmanen.

We present some new results for the asymptotic behavior of the complex parabolic Ginzburg–Landau equation. In particular, we establish that, as the parameter ε tends to 0, vorticity evolves according to motion by mean curvature in Brakke's weak formulation. The only assumption we make is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen.

Reçu le : 2003-03-17
Accepté le : 2003-03-17
Publié le : 2003-04-30

DOI : 10.1016/S1631-073X(03)00167-5

Affiliations des auteurs :

Fabrice Bethuel ^{1, 2} ; Giandomenico Orlandi ³ ; Didier Smets ¹

¹ Laboratoire Jacques-Louis Lions, Université de Paris 6, 4, place Jussieu, BC 187, 75252 Paris, France
² Institut universitaire de France
³ Dipartimento di Informatica, Università di Verona, Strada le Grazie, 37134 Verona, Italy

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Fabrice Bethuel; Giandomenico Orlandi; Didier Smets. Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature. Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 719-723. doi : 10.1016/S1631-073X(03)00167-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00167-5/

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