Uniform estimates for the parabolic Ginzburg-Landau equation

Bethuel, F.  ; Orlandi, G. 

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 219-238

Résumé

We consider complex-valued solutions $u_{\epsilon }$ of the Ginzburg-Landau equation on a smooth bounded simply connected domain $\Omega$ of ${ℝ}^N$, $N\ge 2$, where $\epsilon >0$ is a small parameter. We assume that the Ginzburg-Landau energy $E_{\epsilon }\left(u_{\epsilon }\right)$ verifies the bound (natural in the context) $E_{\epsilon }\left(u_{\epsilon }\right)\le M_0|log\epsilon |$, where $M_0$ is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of $u_{\epsilon }$, as $\epsilon \to 0$, 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.

Détail

DOI : 10.1051/cocv:2002026

Classification : 35K55, 35J60, 58E50, 49J10
Keywords: Ginzburg-Landau, parabolic equations, Hodge-de Rham decomposition, jacobians