Approximation of the Global Attractors of Evolution Equations

这位研究者和他的同事研究了耗散偏微分方程解的整体吸引子和不变流形的计算。全局吸引子的检测是通过解析内插(在复数时间内)而不是直接模拟来实现的。这种方法的性能与时间倒退的解的增长速度有关。为了确定一组具有特定比率的解是否真的是流形，人们进行了计算。对惯性流形的一个连续压缩映射算法进行了改编，使之适用于中心流形和叶层。具体应用于物理模型，包括Camassa-Holm方程、Kuramoto-Sivashinsky方程和Navier-Stokes方程，包括湍流理论中的问题。这个项目涉及描述燃烧、流体流动和湍流等物理现象的各种数学方程。这项工作的一部分是开发一种方法，可靠地确定特定的计算解决方案是代表永久行为还是仅仅代表暂时行为。另一个涉及几何对象的可视化，这些几何对象对应于确定物理系统的最终行为的不同解集。虽然这项工作是针对几个具体的、具有物理意义的方程进行的，但它足够普遍，可以自然地扩展到许多其他类似形式的方程，包括一些模拟天气的方程。

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.