Analysis of Ginzburg-Landau models

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The goal of the proposed research is the rigorous analysis of nonlinear partial differential equations (PDE) and variational problems arising from models of the physical world. While the Ginzburg-Landau (GL) model was initially introduced to study phase transitions in superconductors, over time it has become ubiquitous in the mathematical description of multi-phase systems in physics and materials science and has lead to a wealth of deep mathematical results and new methods. In a Ginzburg-Landau model the state of a physical system is described as a critical point of a given energy functional, and solves a system of nonlinear elliptic PDE. Heuristic analysis suggests that in certain parameter limits, the solutions may develop geometrical singularities (such as vortices, monopoles, or domain walls,) which give the most salient features of the system and characterize its fundamental interactions and dynamics.This research proposal concerns vector-valued Ginzburg-Landau models (as opposed to scalar-valued) introduced in a variety of physical contexts such as high-temperature superconductivity, Bose-Einstein condensation (BEC), nematic liquid crystals, superfluidity, or ferromagnetism, as well as in the study of microstructures in alloys. Each one is a singular perturbation problem, leading to geometrical singularities (either point vortices or transition hypersurfaces,) and we ask: what type of singularities may be observed; how many are there and where; what is the profile of the minimizers near a singularity?An essential ingredient in this study will be the development of Liouville-type theorems, which classify the entire solutions to the appropriate blowup equations in all of space. Very few Liouville theorems are known for systems of elliptic PDE and original methods must be developed to study these systems. The analytical results obtained will give a more complete and reliable understanding of these models and the phenomena they describe, while providing new perspectives on the rich interplay between analysis, geometry, and physics.