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

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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).