Analysis of Ginzburg-Landau models

提出的研究目标是对非线性偏微分方程（PDE）和由物理世界模型引起的变分问题进行严格分析。虽然Ginzburg-Landau （GL）模型最初是为了研究超导体中的相变而引入的，但随着时间的推移，它已经在物理和材料科学中多相系统的数学描述中无处不在，并导致了丰富的深刻的数学结果和新方法。在金兹堡-朗道模型中，物理系统的状态被描述为一个给定能量泛函的临界点，求解了一个非线性椭圆偏微分方程系统。启发式分析表明，在某些参数限制下，解可能会产生几何奇点（如漩涡、单极子或畴壁），这些奇点给出了系统的最显著特征，并表征了其基本相互作用和动力学。

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.