Interaction energy between vortices of vector fields on Riemannian surfaces

Ignat, Radu; Jerrard, Robert L.

doi:10.1016/j.crma.2017.04.004

Partial differential equations/Calculus of variations

Interaction energy between vortices of vector fields on Riemannian surfaces
[Énergie d'interaction entre les tourbillons des champs de vecteurs sur une surface riemannienne]

Présenté par : Haïm Brézis

Ignat, Radu ¹ ; Jerrard, Robert L.²

¹ Institut de Mathématiques de Toulouse, Université Paul-Sabatier, 31062 Toulouse, France
² Department of Mathematics, University of Toronto, Toronto, Ontario, Canada

Comptes Rendus. Mathématique, Tome 355 (2017) no. 5, pp. 515-521.

Résumé
Abstract

Nous étudions un modèle variationnel de type Ginzburg–Landau (dépendant d'un petit paramètre $ε > 0$ ) pour des champs de vecteurs (tangents) sur une surface riemannienne. Lorsque $ε \to 0$ , ces champs de vecteurs auront des points singuliers d'indice non nul, appelés tourbillons. Notre résultat détermine l'énergie d'interaction entre les tourbillons en tant que Γ-limite (au second ordre) pour $ε \to 0$ .

We study a variational Ginzburg–Landau-type model depending on a small parameter $ε > 0$ for (tangent) vector fields on a 2-dimensional Riemannian surface. As $ε \to 0$ , the vector fields tend to be of unit length and will have singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as a Γ-limit (at the second order) as $ε \to 0$ .

Reçu le : 2017-01-23
Accepté le : 2017-04-07
Publié le : 2017-04-20

DOI : 10.1016/j.crma.2017.04.004

Affiliations des auteurs :

Ignat, Radu ¹ ; Jerrard, Robert L. ²

¹ Institut de Mathématiques de Toulouse, Université Paul-Sabatier, 31062 Toulouse, France
² Department of Mathematics, University of Toronto, Toronto, Ontario, Canada

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     title = {Interaction energy between vortices of vector fields on {Riemannian} surfaces},
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     pages = {515--521},
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Ignat, Radu; Jerrard, Robert L. Interaction energy between vortices of vector fields on Riemannian surfaces. Comptes Rendus. Mathématique, Tome 355 (2017) no. 5, pp. 515-521. doi : 10.1016/j.crma.2017.04.004. http://www.numdam.org/articles/10.1016/j.crma.2017.04.004/

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