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Uniform estimates for the parabolic Ginzburg–Landau equation

Published online by Cambridge University Press:  15 August 2002

F. Bethuel  and
G. Orlandi
F. Bethuel
Affiliation:
Analyse Numerique, Université P. et M. Curie, BC 187, 4 place Jussieu 75252 Paris Cedex 05, France; [email protected].
G. Orlandi
Affiliation:
Dipartimento di Informatica, Università di Verona, strada le Grazie, 37134 Verona, Italy.
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Abstract

We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of $\mathbb{R}^N$ , N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy $E_\varepsilon(u_\varepsilon)$ verifies the bound (natural in the context) $E_\varepsilon(u_\varepsilon)\le M_0|\log\varepsilon|$ , where M 0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of uE, as ε → 0, is to establish uniform Lp bounds for the gradient, for some p>1. We review some recent techniques developed in the elliptic case in [7], discuss some variants, and extend the methods to the associated parabolic equation.

Keywords

Ginzburg–Landauparabolic equationsHodge–de RhamdecompositionJacobians.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 219 - 238
Copyright
© EDP Sciences, SMAI, 2002

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