Analysis of Ginzburg-Landau models

拟议的研究的目标是严格分析非线性偏微分方程（PDE）和变分问题所产生的模型的物理世界。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，它在物理学和材料科学中的多相系统的数学描述中变得无处不在，并导致了丰富的深刻的数学结果和新方法。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，并求解非线性椭圆偏微分方程组。 启发式分析表明，在一定的参数限制，解决方案可能会发展的几何奇点（如旋涡，单极，或域壁，），这给系统的最显着的特点，并表征其基本的相互作用和动力学。

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.