Uniform estimates for the parabolic Ginzburg-Landau equation

Bethuel, F.; Orlandi, G.

doi:10.1051/cocv:2002026

Bethuel, F. ; Orlandi, G.

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

Résumé

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

Zbl | 1 citation dans Numdam

DOI : 10.1051/cocv:2002026

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

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     title = {Uniform estimates for the parabolic {Ginzburg-Landau} equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {219--238},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002026},
     zbl = {1078.35013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002026/}
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Bethuel, F.; Orlandi, G. Uniform estimates for the parabolic Ginzburg-Landau equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 219-238. doi : 10.1051/cocv:2002026. http://www.numdam.org/articles/10.1051/cocv:2002026/

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