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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 ε (u)= Ω ε|u| 2 +ε -1 (1-|u| 2 ) 2 dx, where ε>0 and Ω 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:
49J45Optimal control problems involving semicontinuity and convergence; relaxation
35J20Second order elliptic equations, variational methods
58E50Applications of variational methods in infinite-dimensional spaces
82D55Superconductors (statistical mechanics)