In this paper we consider the asymptotic behavior of the Ginzburg-Landau model for superconductivity in 3-d, in various energy regimes. We rigorously derive, through an analysis via Γ-convergence, a reduced model for the vortex density, and deduce a curvature equation for the vortex lines. In a companion paper we describe further applications to superconductivity and superfluidity, such as general expressions for the first critical magnetic field Hc1, and the critical angular velocity of rotating Bose-Einstein condensates.
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