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

凯夫瑞基迪斯9711224研究员和他的合作者，加州大学欧文分校的Edriss Titi，研究了耗散发展偏微分方程解在扰动下的长期行为；他们采取了理论和计算机辅助相结合的方法，并考虑了一些说明性的应用。这些时空扰动的物理动机是发生在具有不同性质的介质中的现象，例如非均匀介质中的反应和扩散(导致具有空间或时空相关系数的偏微分方程组)。它们也可能是空间分布系统上的反馈控制循环的结果。在该项目中，他们感兴趣的是通过大幅度扰动和/或尺度变化效应来维持/利用/规定低维动力学。他们建立的工具是惯性和近似惯性流形的全局机制，以及用于模拟、分叉和分析非线性演化偏微分方程组的科学计算。他们解决的这组新问题需要将这些工具与控制理论中的时间尺度分离方面(例如，闭环系统中惯性或近似惯性流形的持久性)或齐化理论(当PDE中表示介质性质的系数在不同的空间尺度上变化时)的方面进行扩展和结合。这个项目扩展、开发和实现了数学和计算工具，以增强我们研究非均匀条件下的反应和传输过程(由耗散的非线性发展偏微分方程组建模)的能力。在现实的物理环境中，这样的条件更多地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，我们希望保证一定水平的性能)，或者是由于有意设计的复合介质，或者是由于反馈控制(在这种情况下，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，特殊的应用集中在用于多相反应的新型复合催化剂的建模、分析和设计，以及用于空间扩展系统(如流体流动)控制的建模的开发。

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).