Partial Differential Equations
A product estimate for Ginzburg–Landau and application to the gradient-flow
Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 997-1002.

We prove a new inequality for the Jacobian (or vorticity) associated to the Ginzburg–Landau energy in any dimension, and give static and dynamical corollaries. We then present a method to prove convergence of gradient-flows of families of energies which Gamma-converge to a limiting energy, which we apply to establish, thanks to the previous dynamical estimate, the limiting dynamical law of a finite number of vortices for the heat-flow of Ginzburg–Landau in dimension 2, with and without magnetic field.

Nous prouvons une nouvelle inégalite sur le jacobien (ou vorticité) associé à l'énergie de Ginzburg–Landau en dimension quelconque, et en donnons des corollaires statiques et dynamiques. Nous présentons ensuite une méthode pour prouver la convergence de flots-gradient associés à une famille d'énergies qui Gamma-convergent vers une énergie limite, que nous appliquons pour établir à l'aide de l'estimée dynamique précédemment obtenue, la loi limite de la dynamique d'un nombre fini de vortex pour le flot (de la chaleur) de Ginzburg–Landau en dimension 2 avec et sans champ magnétique.

Received:
Published online:
DOI: 10.1016/S1631-073X(03)00224-3
Sandier, Etienne 1; Serfaty, Sylvia 2

1 Mathématiques, Université Paris-12 Val-de-Marne, 61, ave du Général de Gaulle, 94010 Créteil cedex, France
2 Courant Institute of Mathematical Sciences, 251 Mercer st, New York, NY 10012, USA
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Sandier, Etienne; Serfaty, Sylvia. A product estimate for Ginzburg–Landau and application to the gradient-flow. Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 997-1002. doi : 10.1016/S1631-073X(03)00224-3. http://www.numdam.org/articles/10.1016/S1631-073X(03)00224-3/

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