Collaborative Research: Waves and Fronts in Heterogeneous Media

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Numerical simulation of the microscopic details of wave propagation in random media is still beyond the reach of modern computers: a typical propagation distance may be on the order of hundreds of wavelengths and as many correlation lengths of random fluctuations. This necessitates the use of various approximate macroscopic models, of which kinetic equations constitute an important class. However, the passage from microscopic wave equations to large-scale kinetics is a complicated problem in itself. The goal of the first part of the project is to develop new tools and better understanding of kinetic limits, especially in the regime when random media have long range correlations that lead to multiple temporal and spatial scales for various wave phenomena. The second part of the project investigates the qualitative behavior of solutions of reaction-diffusion-advection equations, with the main focus on the effect of a fluid flow. The interaction of the mixing, dynamic, and geometric properties of the underlying flow and the effects of diffusion and reaction will be investigated. The problem becomes especially complex in the situations where the feedback from the reaction process on the fluid flow cannot be ignored. The project addresses the quantitative study of the transport of the energy, momentum, and reactants in a Boussinesq reactive system, other reaction-diffusion systems with feedback, and flame-front propagation in random media.This project carries out mathematical studies of wave propagation in cluttered environments and of reaction-diffusion processes in fluids. These studies are relevant to several branches of science, ranging from biomedical imaging to geophysics, fluid dynamics, and astrophysics. Imaging in a cluttered environment, whether it is a human body, earth interior, or foliage, is inherently unstable because of media complexity. One objective of this project is to develop imaging methods that are less sensitive to unpredictable fluctuations of the clutter. Another area of this project concerns the mathematical description of the effect of a fluid flow on chemical reactions. Turbulent fluid flow plays an important role in many chemical phenomena: it may drastically enhance the rate of reaction, leading to higher efficiency, or, in some situations, extinguish the chemical process. The project will address these issues in simpler mathematical models to illuminate the mechanisms present in the full problem.