Analysis of Defect Measures and Their Applications

本文的主题是凝聚态物理中缺陷的严格数学理论。特别令人感兴趣的是液晶中的缺陷和超导体中的涡流（细丝）。一种是通过分析所谓的缺陷测量来描述这些缺陷的几何性质、拓扑结构和动力学行为。这是一项相当艰巨的任务。对于液晶，必须首先研究一类非线性耦合的Navier-Stokes型方程与演化近似谐波映射系统的非线性耦合。对于超导，研究具有一定非线性效应（Eliashberg-Gorkov方程）的Yang-Mills流动。下一个关键问题是理解解的奇点的发展，如能量集中、尖锐界面和冒泡现象。特别地，我们必须检查谐波和近似谐波图的流动中的奇点，以及金兹堡-朗道方程中漩涡和细丝的动力学。本研究还将对经典流体力学中出现的其他问题提供见解。各种自然现象可以用某些偏微分方程的解来描述。通常，解的奇异行为不仅揭示了它们所描述的问题的各个方面，而且还揭示了问题的基本特征。后者对于技术和工业应用非常重要。例如，II型超导体（高温超导体）的特点是存在一定的涡流和细丝的晶格结构。控制这些涡流和细丝的动力学是这些高科技材料适用性的核心问题之一。因此，这是一个持续关注的非常基本的问题。尽管做出了许多认真的努力，但迄今为止，解决这类问题的数学方法很少。本文提出了一种新的理论方法，经过初步分析，这种方法对材料科学中出现的一类问题是有用的。现代技术需要高性能、高能效和高精度。液晶和超导体都属于这一类。这项研究将产生新的和深刻的定性和定量信息关于这些迷人的材料，除了有自己内在的数学重要性和兴趣。

9706862 Lin The main theme of this proposal is the rigorous mathematical theory of defects in condensed matter physics. Of special interest are defects in liquid crystals and vortices (filaments) in superconductors. One describes the geometrical properties, topological structures, and dynamical behavior of these defects by analyzing the so-called defect measures. It is a rather formidable task. For liquid crystals one has to study first a certain nonlinear coupling of equations of Navier-Stokes type with those of evolutionary approximate harmonic map systems. For superconductivity, one studies then the flow of Yang-Mills with certain nonlinear effects (Eliashberg-Gorkov equations). The next key issues are understanding the development of singularities of solutions, such as energy concentrations, sharp interfaces, and bubbling phenomena. In particular, one must examine singularities in the flow of harmonic and approximate harmonic maps and the dynamics of vortices and filaments in Ginzburg-Landau equations. This study should also give insight into other problems that arise in classical fluid dynamics. Various natural phenomena can be described by solutions of certain partial differential equations. Often singular behavior of solutions reveal not only facets but also essential characteristics of the problems they describe. The latter is very important for technological and industrial applications. For example, the type II superconductors (high-temperature super-conductors) are characterized by the existence of a certain lattice structure of vortices and filaments. To control the dynamics of these vortices and filaments is one of the central issues for the applicability of these high-tech materials. It is, therefore, a very basic problem of continuing interest. Despite many serious efforts, very few mathematical methods exist so far to tackle such problems. The present proposal presents a new and novel theoretical approach which has already been shown, by preliminary analysis, to be useful for a class of problems arising in material science. Modern technology needs high performance, high energy efficiency, and high accuracy. Both liquid crystals and superconductors are in such a category. This study will yield new and insightful qualitative and quantitative information regarding these fascinating materials, in addition to having its own intrinsic mathematical importance and interest.