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hodge–de rhamdecomposition

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Uniform estimates for the parabolic Ginzburg–Landau equation
F. Bethuel, G. Orlandi
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 8 / 2002

Published online by Cambridge University Press:

15 August 2002, pp. 219-238

Print publication:

2002
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We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of $\mathbb{R}^N$ , N ≥ 2, where ε > 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 uE, as ε → 0, is to establish uniform Lp 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.