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Zbl 1065.49011
Sandier, Etienne; Serfaty, Sylvia
Gamma-convergence of gradient flows with applications to Ginzburg-Landau.
(English)
[J] Commun. Pure Appl. Math. 57, No. 12, 1627-1672 (2004). ISSN 0010-3640

This paper is devoted to the study of gradient flows of the Ginzburg-Landau energy functional $E_\varepsilon (u)=\int_\Omega \left[\varepsilon \vert \nabla u\vert ^2+\varepsilon^{-1}(1-\vert u\vert ^2)^2\right]dx$, where $\varepsilon>0$ and $\Omega\subset\Bbb R^2$ is a smooth, bounded and simply connected domain. The authors establish several properties of gradient flows from the viewpoint of the Gamma-convergence theory. One of the first results of the paper provides lower-bound criteria to deduce an appropriate convergence property. Using this result the authors prove the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy. The proofs combine powerful elliptic estimates and adequate minimization methods.
[Vicenţiu D. Rădulescu (Craiova)]
MSC 2000:
*49J45 Optimal control problems inv. semicontinuity and convergence
35J20 Second order elliptic equations, variational methods
58E50 Appl. of variational methods in infinite-dimensional spaces
82D55 Superconductors

Keywords: Gamma-convergence; Ginzburg-Landau energy functional; asymptotic analysis; gradient flow

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