A branched transport limit of the Ginzburg-Landau functional

Abstract

We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via Γ-convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.

Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys

“…One obtains mesoscopic phase diagrams which characterize the different regimes of material behavior and the qualitative properties of the microstructure [41,10,38,39]. The techniques developed for singularly-perturbed functionals modeling martensitic microstructures have proven useful also in the study of a variety of other physical problems, such as for example magnetic microstructures [17,19,40], flux tubes in superconductors [18,23,24], diblock copolymers [16], wrinkling in thin elastic films [36,8,7], and compliance minimization [44].…”

On the well-posedness of branched transportation

“…The models described above can be used and generalized to describe a variety of problems related to branched transportation: for instance, one can study the mailing problem [2] (for which the first stability result was proved in [18]), the urban planning model [8], including two different regimes of transportation, or the recent multi-material transport problem [32,33], allowing simultaneous transportation of different goods or commodities. Recently, shape optimization problems related to the functional (1.1) were analysed in [41,11] and similar branching structures are observed in superconductivity models and for minimizers of Ginzburg-Landau type functionals, see for instance [27,14,15,16,20].…”

Self-similar minimizers of a branched transport functional

“…However, it is usually very hard to go beyond scaling laws [12,21,7,11]. In some cases, reduced models have been derived [15,10,9] but so far the best results concerning the minimizers are local energy bounds leading to the proof of asymptotic self-similarity [8,18]. Our result is thus the first complete characterization of a minimizer in this context.…”

“…The variational problem (1.2) may be seen as a two dimensional (one for time and one for space) analog of the three dimensional (one for time and two for space) problem derived in [10] as a reduced model for the description of branching in type-I superconductors in the regime of very small applied external field. We refer the reader to [10] for more precise physical motivations and references. In this regime, the natural Dirichlet conditions appearing are µ ± = dx [−1/2, 1/2].…”