Approximation of the Global Attractors of Evolution Equations

研究者和他的同事研究了耗散偏微分方程的整体吸引子和不变流形的计算。 全局吸引子的检测通过插值的分析（在复杂的时间），而不是直接模拟。 这种方法的性能是连接到解决方案的增长率在时间上落后。 计算是为了确定一组具有特殊特征的解是否是流形。 本文将惯性流形的一个成功的压缩映射算法推广到中心流形和叶理。 具体的应用是madeto物理模型，包括Camassa-Holm，Kuramoto-Sivashinsky和Navier-Stokes方程，包括湍流理论的问题。 这个项目涉及各种数学方程，描述了燃烧、流体流动和湍流等物理现象。 部分工作是开发一种方法来可靠地确定一个特定的计算解决方案是否代表永久或仅仅是暂时的行为。 另一个关注的几何对象的可视化对应的区别集的解决方案，这决定了最终的物理系统的行为。 虽然这项工作是针对几个具体的、物理上重要的方程进行的，但它足够普遍，可以以一种自然的方式扩展到许多类似形式的其他方程，包括一些天气模型。

The investigator and his colleague atudy the computation ofglobal attractors and invariant manifolds for dissipative partialdifferential equations. Global attractors are detected byinterpolation of analytic (in complex time) rather than directsimulations. The performance of this approach is connected tothe growth rate of solutions backward in time. Computations arecarried out to determine if a set of solutions with a particularrate is in fact a manifold. Adaptations of a succesfulcontraction mapping algorthm for inertial manifolds are made forcenter manifolds and foliations. Specific applications are madeto physical models including the Camassa-Holm,Kuramoto-Sivashinsky and Navier-Stokes equations, includingissues from turbulence theory. This project concerns a variety of mathematical equationsdescribing such physical phenomena as combustion, fluid flow, andturbulence. Part of the effort is to develop a method toreliably determine whether a particular computed solutionrepresents permanent or merely temporary behavior. Anotherconcerns the visualization of geometric objects corresponding todistinguished sets of solutions, which determine the ultimatebehavior of the physical system. Though the work is carried outfor several specific, physically significant equations, it isgeneral enough to extend, in a natural way, to many others of asimilar form, including some modeling the weather.