Analysis of Ginzburg-Landau models

这项研究的目标是对非线性偏微分方程(PDE)和物理世界模型产生的变分问题进行严格的分析。虽然Ginzburg-Landau(GL)模型最初是用来研究超导体中的相变的，但随着时间的推移，它在物理和材料科学中的多相系统的数学描述中变得无处不在，并产生了丰富的深刻的数学结果和新的方法。在Ginzburg-Landau模型中，物理系统的状态被描述为给定能量泛函的临界点，并求解非线性椭圆型偏微分方程组。启发式分析表明，在某些参数限制下，解可能会产生几何奇异性(如涡旋、单极子或磁区壁)，这些奇点给出了系统最显著的特征，并表征了其基本的相互作用和动力学。 这一研究建议涉及在各种物理环境中引入的矢量值Ginzburg-Landau模型(与标量值相反)，如高温超导、玻色-爱因斯坦凝聚(BEC)、向列相液晶、超流性或铁磁性，以及在合金微结构研究中的应用。每个问题都是一个奇异摄动问题，导致几何奇点(点涡旋或过渡超曲面)，我们问：可以观察到什么类型的奇点；有多少个奇点？奇点附近的极小点的轮廓是什么？ 这项研究的一个基本组成部分将是Liouville类型定理的发展，它将所有空间中适当的爆破方程的全部解分类。已知的椭圆型偏微分方程组的Liouville定理很少，必须发展独创的方法来研究这些系统。所获得的分析结果将使我们对这些模型及其所描述的现象有一个更全面、更可靠的理解，同时也为分析、几何和物理之间的丰富相互作用提供了新的视角。

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.