Uniform estimates for the parabolic Ginzburg–Landau equation

F. Bethuel; G. Orlandi

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

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We consider complex-valued solutions uE 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 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.

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 (2010): 219-238. <http://eudml.org/doc/90647>.

@article{Bethuel2010,
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. },
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},
month = {3},
pages = {219-238},
publisher = {EDP Sciences},
title = {Uniform estimates for the parabolic Ginzburg–Landau equation},
url = {http://eudml.org/doc/90647},
volume = {8},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 219
EP - 238
AB - 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.
LA - eng
KW - Ginzburg–Landau; parabolic equations; Hodge–de Rham decomposition; Jacobians.; Hodge-de Rham decomposition; Jacobians
UR - http://eudml.org/doc/90647
ER -

References

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