This project carries out mathematical studies of physical systems in heterogeneous environments. Such systems are ubiquitous in nature, and include examples as diverse as light propagation through the atmosphere, seismic waves in the Earth following an earthquake, stock market fluctuations, and medical imaging problems. The mathematical modeling of such problems involves partial differential equations with highly oscillatory coefficients. Typically, such problems involve a multitude of temporal and spatial scales: a typical propagation distance may be of the order of hundreds or thousands of wavelengths and as many correlation lengths of random fluctuations. Numerical simulation of the microscopic details of the solutions is beyond the reach even for the most powerful modern computers. For the prediction and understanding of behavior of these multi-scale system, it is imperative to use various approximate macroscopic effective models that preserve the salient features of the processes without keeping track of all microscopic details. The overarching goal of the project is to develop a better understanding of the validity of such macroscopic models.
The goal of the first part of the project is to develop new tools and better understanding of the effective limits in wave propagation, with the focus on random media with long-range correlations that lead to multiple temporal and spatial scales for various physical phenomena. The second part of the proposal investigates the qualitative behavior of the solutions of reaction-diffusion equations with stochastic perturbations. Stochastic fluctuations and heterogeneities play a significant effect on front propagation properties, leading to new types of solutions and asymptotic behavior. One goal of the project will be to understand the connection between the stochastic reaction-diffusion problems and the recent theory of the regularity structures.
