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Gamma-convergence of gradient flows with applications to Ginzburg-Landau. (English) Zbl 1065.49011
This paper is devoted to the study of gradient flows of the Ginzburg-Landau energy functional $$E_\varepsilon (u)=\int_\Omega \left[\varepsilon | \nabla u| ^2+\varepsilon^{-1}(1-| u| ^2)^2\right]dx$$, where $$\varepsilon>0$$ and $$\Omega\subset\mathbb R^2$$ 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.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 35J20 Variational methods for second-order elliptic equations 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences 82D55 Statistical mechanical studies of superconductors
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