Waves, Particle Transport and Fronts in Heterogeneous Media

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AbstractNumerical simulation of the microscopic details of wave propagation in random media is still beyond reach of modern computers: a typical propagation distance may be of the order of hundreds of wavelengths and as many correlation lengths of random fluctuations. This necessitates the use of various approximate macroscopic effective models in practice. The passage from the stochastic microscopic equations modeling a particular physical system to the large-scale model is a highly non-trivial problem in itself. The goal of the first part of the project is to develop new tools and better understanding of such effective limits, especially in the regime when random media have long range correlations that lead to multiple temporal and spatial scales for various physical phenomena. The second part of the project investigates the qualitative behavior of solutions of reaction-diffusion equations, in particular, those that arise in stochastic particle systems, such as branching Brownian motion, and other Brunet-Derrida systems. Another set of problems involves reaction-diffusion equations that arise in biology, such as chemotactic models, and ecology involving population dynamics in heterogenous environments. This project carries out mathematical studies of physical systems, and wave propagation in cluttered environments and of reaction-diffusion processes. These studies are relevant to several branches of science, ranging from biomedical imaging questions to geophysics, fluid dynamics, and biology. 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 mixing effects of a fluid flow on chemical and biological reactions. Turbulent fluid flow plays an important role in many reaction 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.