Analysis of Defect Measures and Their Applications

9706862林这项提议的主题是凝聚态物理中缺陷的严谨数学理论。特别令人感兴趣的是液晶中的缺陷和超导体中的涡旋(细丝)。一种是通过分析所谓的缺陷度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，首先必须研究Navier-Stokes方程与演化近似调和映射系统之间的某种非线性耦合。对于超导性，人们接着研究具有某些非线性效应的杨-米尔斯流(Eliashberg-Gorkov方程)。下一个关键问题是理解解的奇异性的发展，例如能量集中、尖锐的界面和起泡现象。特别是，我们必须研究Ginzburg-Landau方程中调和映射和近似调和映射流动的奇异性，以及涡旋和细丝的动力学。这项研究还应该对经典流体力学中出现的其他问题有更深入的了解。各种自然现象可以用某些偏微分方程解来描述。通常，解决方案的奇异行为不仅揭示了问题的各个方面，而且揭示了它们所描述的问题的本质特征。后者对于技术和工业应用非常重要。例如，第二类超导体(高温超导体)的特征是存在一定的涡旋和细丝的晶格结构。控制这些涡旋和细丝的动力学是这些高科技材料能否应用的核心问题之一。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，但到目前为止，解决这类问题的数学方法很少。本建议提出了一种新的、新颖的理论方法，初步分析表明，这种方法对材料科学中出现的一类问题是有用的。现代技术要求高性能、高能效、高精度。液晶和超导体都属于这一类。这项研究将产生关于这些引人入胜的材料的新的和有洞察力的定性和定量信息，此外，它还具有自己固有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.