Uniform estimates for the parabolic Ginzburg-Landau equation
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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.

DOI: 10.1051/cocv:2002026
Classification: 35K55,  35J60,  58E50,  49J10
Keywords: Ginzburg-Landau, parabolic equations, Hodge-de Rham decomposition, jacobians
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title = {Uniform estimates for the parabolic {Ginzburg-Landau} equation},
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Bethuel, F.; Orlandi, G. Uniform estimates for the parabolic Ginzburg-Landau equation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 219-238. doi : 10.1051/cocv:2002026. http://archive.numdam.org/articles/10.1051/cocv:2002026/

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