Uniform estimates for the parabolic Ginzburg–Landau equation

¹ Analyse Numerique, Université P. et M. Curie, BC 187, 4 place Jussieu 75252 Paris Cedex 05, France; bethuel@ann.jussieu.fr.
² Dipartimento di Informatica, Università di Verona, strada le Grazie, 37134 Verona, Italy.

Received: 17 December 2001
Revised: 25 January 2002

Abstract

We consider complex-valued solutions u_E 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 u_E, as ε → 0, is to establish uniform L^p 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.