Uniform estimates for the parabolic Ginzburg–Landau equation
Analyse Numerique, Université P.
et M. Curie, BC 187, 4 place Jussieu 75252 Paris Cedex 05, France; email@example.com.
2 Dipartimento di Informatica, Università di Verona, strada le Grazie, 37134 Verona, Italy.
Revised: 25 January 2002
We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of , N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy verifies the bound (natural in the context) , where M0 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 , discuss some variants, and extend the methods to the associated parabolic equation.
Mathematics Subject Classification: 35K55 / 35J60 / 58E50 / 49J10
Key words: Ginzburg–Landau / parabolic equations / Hodge–de Rham decomposition / Jacobians.
© EDP Sciences, SMAI, 2002