Many physical systems described by partial differential equations (PDEs) are influenced by microscopic variations that may be characterized as random. A fundamental scientific and mathematical problem is to understand how these random variations influence the system at a macroscopic level. The investigator will study partial differential equations with random coefficients. In some regimes, the macroscopic system is well-approximated by the solution to a simpler "effective equation." In other regimes, however, the macroscopic effective equation is insufficient to characterize the system's behavior because random microscopic variations produce residual fluctuations in the macroscopic system. In this case, it is important to understand the deviations from the mean behavior. The first part of the project is to characterize the random fluctuations of the wave-like and pulse-like solutions to reaction-diffusion equations and some related equations. Because of the random fluctuation in the coefficients, the solution is random, and it is important to understand how the statistical character of the solution depends on statistical properties of the medium. The second goal is to study random fluctuations of solutions to higher-dimensional models for interface motion. The third goal of the project is to study fluctuations of solutions to elliptic equations with random coefficients.
Understanding the effect of random microstructure on a macroscopic mathematical model is an important problem in many scientific and engineering applications. How can one quantify uncertainty in predictions of a model when only some statistical properties of the model parameters are known? The analytical techniques developed through this project will have an impact in the area of uncertainty quantification for PDE-based mathematical models. The specific equations to be studied include reaction-diffusion equations which arise in models of phenomena in turbulent combustion, population ecology, and neuroscience, for example. The project also will consider elliptic equations which arise in applications such as hydrology and materials science. In these applications, material properties like permeability or electrical conductivity vary randomly and only some statistical properties of these parameters are known.
