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We present a method to prove convergence of gradient flows of families of energies that -converge to a limiting energy. It provides lower-bound criteria to obtain the convergence that correspond to a sort of C 1 -order -convergence of functionals. We then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy in dimension 2, retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field.

We introduce a "Coulombian renormalized energy" W which is a logarithmic type of interaction between points in the plane, computed by a "renormalization." We prove various of its properties, such as the existence of minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique minimizer. Its minimization in general remains open.Our motivation is the study of minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields H c 1 and H c 2 . In that regime, minimizing configurations exhibit densely packed triangular vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, W as a -limit of the Ginzburg-Landau energy. More precisely we show that the vortices of minimizers of Ginzburg-Landau, blown-up at a suitable scale, converge to minimizers of W , thus providing a first rigorous hint at the Abrikosov lattice. This is a next order effect compared to the mean-field type results we previously established.The derivation of W uses energy methods: the framework of -convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem, that we introduce.

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case β = ∞ corresponds to "weighted Fekete sets" and also falls within our analysis.It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni [BZ].By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" W , a Coulombian interaction for points in the plane introduced in [SS1], which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When β → ∞, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of W . In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W , which are conjectured to be "Abrikosov" triangular lattices.

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A rigorous derivation of free-boundary problem arising in superconductivity Annales scientifiques de l'É.N.S. 4 e série, tome 33, n o 4 (2000), p. 561-592 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 2000, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

We study the Ginzburg–Landau energy of superconductors with high κ, put in a prescribed external field h ex , for h ex varying between the two critical fields Hc1 and Hc3. As κ → +∞, we give the leading term in the asymptotic expansion of the minimal energy and show that energy minimizers have vortices whose density tends to be uniform and equal to h ex .

We study the statistical mechanics of a one-dimensional log gas or β-ensemble with general potential and arbitrary β, the inverse of temperature, according to the method we introduced for two-dimensional Coulomb gases in Sandier and Serfaty (Ann Probab, 2014). Such ensembles correspond to random matrix models in some particular cases. The formal limit β = ∞ corresponds to "weighted Fekete sets" and is also treated. We introduce a one-dimensional version of the "renormalized energy" of Sandier and Serfaty (Commun Math Phys 313(3):635-743, 2012), measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. We show that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian we connect the full statistical mechanics problem to this renormalized energy W , and this allows us to obtain new results on the distribution of the points at the microscopic scale: in