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Uniform estimates for the parabolic Ginzburg–Landau equation
Published online by Cambridge University Press: 15 August 2002
Abstract
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 M0 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.
- Type
- Research Article
- Information
- ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 219 - 238
- Copyright
- © EDP Sciences, SMAI, 2002
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