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.
