Uniform estimates for the parabolic Ginzburg-Landau equation. A tribute to J. L. Lions.

We consider complex-valued solutions ue of the Ginzburg-Landau equation on a smooth bounded simply connected domain W of RN, N ³ 2, where e > 0 is a small parameter. We assume that the Ginzburg-Landau energy Ee(ue) verifies the bound (natural in the context) Ee(ue) M|log e|, 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 e® 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