Approximation of the Global Attractors of Evolution Equations

进化方程全局吸引子的近似

• 批准号: 0074460
• 负责人: Michael Jolly
• 金额: $ 19.11万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074460&HistoricalAwards=false
• 关键词: Approximation; Global; Attractors; Evolution; Equations

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.

研究者和他的同事研究了耗散偏微分方程的全局吸引子和不变流形的计算。 全局吸引子是通过解析插值（在复杂时间内）而不是直接模拟来检测的。 这种方法的性能与时间上解决方案的增长率有关。 进行计算以确定具有特定速率的一组解实际上是否是流形。 针对中心流形和叶状结构，对惯性流形的成功收缩映射算法进行了改编。 对物理模型进行了具体应用，包括 Camassa-Holm、Kuramoto-Sivashinsky 和 ​​Navier-Stokes 方程，包括来自湍流理论的问题。 该项目涉及描述燃烧、流体流动和湍流等物理现象的各种数学方程。 部分工作是开发一种方法来可靠地确定特定的计算解决方案是否代表永久行为或仅代表临时行为。 另一个问题涉及与不同解集相对应的几何对象的可视化，这些解集决定了物理系统的最终行为。 尽管这项工作是针对几个特定的​​、具有物理意义的方程进行的，但它具有足够的通用性，可以自然地扩展到许多其他类似形式，包括一些天气建模。