# Uniform estimates for the parabolic Ginzburg-Landau equation

created on 12 Dec 2002

[BibTeX]

Published Paper

Inserted: 12 dec 2002

Journal: ESAIM, C.O.C.V.
Number: 8
Pages: 219-238
Year: 2002

Abstract:

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