The Kinetic Theory of Waves and Reactive-Diffusive Fronts

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Numerical simulation of the microscopic description 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 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 two-fold: on one hand, to develop new tools and better understanding of kinetic limits, and second, to consider the applications of kinetic methods to the inverse problems of wave propagation, finding sources and scatterers in a cluttered environment. 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. We will investigate the interaction of the mixing, dynamic, and geometric properties of the underlying flow and the effects of diffusion and reaction. 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 the reactants in a Boussinesq reactive system.brbrThis project carries out mathematical studies of wave propagation in complex media and of reaction-diffusion equations. The mathematical models are relevant to several branches of science, ranging from biomedical imaging questions 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. An objective of this project is to develop imaging methods that are less sensitive to unpredictable fluctuations of the clutter. We will strive to understand the universality and the limits of applicability of macroscopic models and develop inversion algorithms that arise from the macroscopic rather than detailed microscopic models and that are therefore inherently more stable with respect to fluctuations of the environment. 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 reaction phenomena: it may drastically enhance the rate of reaction, leading to higher efficiency, or, in some situations, extinguish the chemical process. The mathematical theory is far from complete, due to the inherent complexity and richness of the phenomena. The project will address these issues in simpler mathematical models to illuminate the mechanisms present in the full problem.