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.

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.

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