Theory and Applications for Infinite Dimensional Dynamical Systems

摘要：本项目将发展Banach空间中半流的不变流形和叶形的几何理论，并重点研究如何将其应用于重要的科学领域。特别令人感兴趣的是，如何在知道在某种意义上存在好的近似的情况下，确定这些结构的存在。将讨论的应用包括晶格动力系统的分析，流体模型中调制波的存在，超导材料中漩涡的形式和动力学，材料相边界的形式和动力学，包括驱动力中随机扰动的形式和动力学，以及良好计算算法的发展。由于大多数物理系统是高度复杂的，通常不可能解出用来描述和预测相关物理现象的方程。即使有可能解出这些方程中的一个，测量的不精确和我们对实际物理定律的不完善的理解也可能使解的意义受到质疑。这个项目将发展一种理论，据此，在某些条件下，人们可以描述解的性质，并保证物理系统按照这种描述以合理的精度运行。将特别注意描述材料科学现象的方程，包括相变和超导性。

PI: Peter Bates, Brigham Young UniversityCo-PI: Keining Lu, Brigham Young UniversityDMS-0200961Abstract:This project will develop the geometrical theory of invariant manifolds and foliations for semiflows in Banach space with emphasis on how this can be applied to important areas of science. Of particular interest are how one may establish the existence of these structures knowing that good approximations exist in some sense. Applications which will be addressed include analysis of lattice dynamical systems, the existence of modulated waves in fluid models, the form and dynamics of vortices in superconducting materials, the form and dynamics of material phase boundaries, including those with random perturbations in the driving forces, and the development of good computational algorithms.Since most physical systems are highly complex, it is usually impossible to solve the equations which are proposed to describe and predict the relevant physical phenomena. Even if it were possible to solve one of these equations, the imprecision in measurement and our imperfect understanding of the actual physical laws may call into question the meaning of the solutions. This project will develop a theory whereby, under certain conditions, one can describe the nature of solutions and guarantee that the physical system behaves according to this description to a reasonable degree of accuracy. Special attention will be given to equations describing phenomena in material science, including phase transitions and superconductivity.