Evolution PDEs in Inhomogeneous Media: Low-Dimensional Dynamics, Computation and Applications

非均匀介质中的演化偏微分方程：低维动力学、计算和应用

• 批准号: 9711224
• 负责人: Yannis Kevrekidis
• 金额: $ 21.4万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-15 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9711224&HistoricalAwards=false
• 关键词: Evolution; PDEs; Inhomogeneous; Media; Low

Kevrekidis 9711224 The investigator and his collaborator Edriss Titi of the University of California, Irvine, study the long-time behavior of solutions to dissipative evolution partial differential equations under perturbations; they undertake a combined theoretical and computer-assisted approach, with a number of illustrative applications in mind. These spatiotemporal perturbations are motivated physically by phenomena occurring in media with varying properties, such as reaction and diffusion in inhomogeneous media (leading to PDEs with spatially or spatiotemporally dependent coefficients). They may also be the result of a feedback control loop on a spatially distributed system. In the project, they are interested in maintaining / exploiting / prescribing low-dimensional dynamics through large amplitude perturbations and/or scale variation effects. The tools they build upon are the global machinery of inertial and approximate inertial manifolds, as well as scientific computing for the simulation and the bifurcation and stability analysis of nonlinear evolution PDEs. The new set of questions they address requires the extension and combination of these tools with aspects of separation of time scales in control theory (e.g. persistence of inertial or approximate inertial manifolds in closed loop systems) or in homogenization theory (when coefficients in the PDE representing properties of the medium vary on disparate spatial scales). This project extends, develops and implements mathematical and computational tools that enhance our ability to study reaction and transport processes (modeled by dissipative nonlinear evolution partial differential equations) under inhomogeneous conditions. Such conditions constitute more the rule than the exception under realistic physical circumstances, whether due to imperfections in the process (in which case we want to guarantee a certain level of performance) or due to intentional design of composite media, or to fe edback control (where we attempt to optimize a process, like the selectivity of a chemical reaction). The method and algorithm development part of the project is applicable to a wide class of such systems. The particular applications, however, focus on the modeling, analysis and design of novel composite catalysts for heterogeneous reactions, and on the exploitation of modeling for the control of spatially extended systems (such as fluid flows).

Kevrekidis 9711224 研究者和他的合作者，来自加州大学欧文分校的Edriss Titi，研究了扰动下耗散演化偏微分方程解的长期行为;他们采用了理论和计算机辅助相结合的方法，并考虑了一些示例性的应用。 这些时空扰动的物理动机的现象发生在介质中的不同属性，如反应和扩散在非均匀介质（导致偏微分方程的空间或时空相关系数）。 它们也可能是空间分布系统上的反馈控制回路的结果。 在该项目中，他们有兴趣通过大幅度扰动和/或尺度变化效应来维持/利用/规定低维动力学。 他们建立的工具是全球机械的惯性和近似惯性流形，以及科学计算的模拟和非线性演化偏微分方程的分叉和稳定性分析。 他们解决的一组新的问题需要这些工具的扩展和组合方面的时间尺度分离控制理论（例如，惯性或近似惯性流形的闭环系统的持久性）或均匀化理论（当系数在PDE中表示介质的属性在不同的空间尺度上变化）。 该项目扩展，开发和实施数学和计算工具，提高我们在非均匀条件下研究反应和运输过程（由耗散非线性演化偏微分方程建模）的能力。 在现实的物理环境下，这些条件构成了规则而不是例外，无论是由于过程中的缺陷（在这种情况下，我们希望保证一定的性能水平），还是由于复合介质的故意设计，或者反馈控制（我们试图优化过程，如化学反应的选择性）。 该项目的方法和算法开发部分适用于广泛的此类系统。 然而，特定的应用程序，集中在非均相反应的新型复合催化剂的建模，分析和设计，并在空间扩展系统（如流体流动）的控制建模的开发。