Uniform estimates for the parabolic Ginzburg–Landau equation

Abstract

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 [7], discuss some variants, and extend the methods to the associated parabolic equation.

Locations

• ESAIM Control Optimisation and Calculus of Variations - View - PDF
• Springer Link (Chiba Institute of Technology) - View - PDF
• French digital mathematics library (Numdam) - View - PDF