A branched transport limit of the Ginzburg-Landau functional

Sergio Conti; Michael Goldman; Felix Otto; Sylvia Serfaty

doi:10.5802/jep.72

A branched transport limit of the Ginzburg-Landau functional
[Dérivation d’une fonctionnelle de type transport branché à partir du modèle de Ginzburg-Landau]

Sergio Conti ¹ ; Michael Goldman ² ; Felix Otto ³ ; Sylvia Serfaty ⁴

¹ Institut für Angewandte Mathematik, Universität Bonn Endenicher Allee 60, 53115 Bonn, Germany
² LJLL, Université Paris Diderot, CNRS, UMR 7598 Bât. Sophie Germain, 75205 Paris Cedex 13, France
³ Max Planck Institute for Mathematics in the Sciences Inselstraße 22, 04103 Leipzig, Germany
⁴ Courant Institute, NYU 251 Mercer Street New York, N.Y. 10012-1185, USA and Institut Universitaire de France & UPMC 4, place Jussieu, 75252 Paris cedex 05, France

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 317-375.

Résumé
Abstract

Nous étudions le modèle de Ginzburg-Landau pour les supraconducteurs de type I dans le régime de faible champ magnétique extérieur. Nous montrons qu’asymptotiquement, le flux magnétique se concentre sur des structures unidimensionnelles minimisant une fonctionnelle de type transport branché. Nous obtenons cette fonctionnelle simplifiée par $Γ$ -convergence à partir du modèle de Ginzburg-Landau complet. Ceci permet d’obtenir une compréhension fine des différentes échelles mises en jeu.

We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via $Γ$ -convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.

Reçu le : 2017-04-04
Accepté le : 2018-02-12
Publié le : 2018-04-08

MR Zbl

DOI : 10.5802/jep.72

Classification : 35Q56, 49S05, 82D55, 49J10, 49S05
Keywords: Gamma convergence, Ginzburg-Landau, branched transportation, pattern formation, type-I superconductors
Mot clés : Gamma convergence, Ginzburg-Landau, transport branché, supraconducteurs de type I

Affiliations des auteurs :

Sergio Conti ¹ ; Michael Goldman ² ; Felix Otto ³ ; Sylvia Serfaty ⁴

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A branched transport limit of {the~Ginzburg-Landau} functional},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     publisher = {\'Ecole polytechnique},
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     year = {2018},
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     zbl = {06988582},
     mrnumber = {3795445},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.72/}
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Sergio Conti; Michael Goldman; Felix Otto; Sylvia Serfaty. A branched transport limit of the Ginzburg-Landau functional. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 317-375. doi : 10.5802/jep.72. https://jep.centre-mersenne.org/articles/10.5802/jep.72/

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