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

研究者和他的合作者普林斯顿大学的I. Kevrekidis研究了扰动下耗散演化偏微分方程解的长时间行为；他们采用理论和计算机辅助相结合的方法，并考虑到一些说明性应用。这些时空扰动是由具有不同性质的介质中发生的现象引起的，例如非均匀介质中的反应和扩散（导致偏微分方程具有空间或时空依赖系数）。它们也可能是空间分布系统上的反馈控制回路的结果。在这个项目中，他们感兴趣的是通过大振幅扰动和/或尺度变化效应来维持/开发/规定低维动态。他们所建立的工具是惯性和近似惯性流形的全局机制，以及用于非线性演化偏微分方程的模拟、分岔和稳定性分析的科学计算。他们解决的一系列新问题需要将这些工具与控制理论（例如，闭环系统中惯性或近似惯性流形的持久性）或均匀化理论（当PDE中的系数表示介质的性质在不同的空间尺度上变化时）中时间尺度分离的方面进行扩展和组合。该项目扩展、开发和实现了数学和计算工具，以增强我们在非齐次条件下研究反应和输运过程（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，这些条件更像是规则，而不是例外，无论是由于过程中的不完善（在这种情况下，我们希望保证一定水平的性能），还是由于复合介质的有意设计，还是由于反馈控制（当我们试图优化过程时，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，特定的应用侧重于非均相反应的新型复合催化剂的建模、分析和设计，以及对空间扩展系统（如流体流动）控制的建模开发。

Titi 9706964 The investigator and his collaborator I. Kevrekidis of Princeton University 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 feedback control (whe re 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).