Uniform estimates for the parabolic Ginzburg–Landau equation

F. Bethuel; G. Orlandi

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 219-238
  • ISSN: 1292-8119

Abstract

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We consider complex-valued solutions u ε of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of ℝ N , N ≥ 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 ) ≤ 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 → 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.

How to cite

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Bethuel, F., and Orlandi, G.. "Uniform estimates for the parabolic Ginzburg–Landau equation." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 219-238. <http://eudml.org/doc/246070>.

@article{Bethuel2002,
abstract = {We consider complex-valued solutions $u_\varepsilon $ of the Ginzburg–Landau equation on a smooth bounded simply connected domain $\Omega $ of $\mathbb \{R\}^N$, $N \ge 2$, where $\varepsilon &gt; 0$ 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 $\varepsilon \rightarrow 0$, is to establish uniform $L^p$ bounds for the gradient, for some $p&gt;1$. We review some recent techniques developed in the elliptic case in [7], discuss some variants, and extend the methods to the associated parabolic equation.},
author = {Bethuel, F., Orlandi, G.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Ginzburg–Landau; parabolic equations; Hodge–de Rham decomposition; jacobians; Hodge-de Rham decomposition; Jacobians},
language = {eng},
pages = {219-238},
publisher = {EDP-Sciences},
title = {Uniform estimates for the parabolic Ginzburg–Landau equation},
url = {http://eudml.org/doc/246070},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Bethuel, F.
AU - Orlandi, G.
TI - Uniform estimates for the parabolic Ginzburg–Landau equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 219
EP - 238
AB - We consider complex-valued solutions $u_\varepsilon $ of the Ginzburg–Landau equation on a smooth bounded simply connected domain $\Omega $ of $\mathbb {R}^N$, $N \ge 2$, where $\varepsilon &gt; 0$ 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 $\varepsilon \rightarrow 0$, is to establish uniform $L^p$ bounds for the gradient, for some $p&gt;1$. We review some recent techniques developed in the elliptic case in [7], discuss some variants, and extend the methods to the associated parabolic equation.
LA - eng
KW - Ginzburg–Landau; parabolic equations; Hodge–de Rham decomposition; jacobians; Hodge-de Rham decomposition; Jacobians
UR - http://eudml.org/doc/246070
ER -

References

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