Long Time Behavior for Partial Differential Equations in Random Media

Stanford University, Stanford CA

Investigators

Abstract

This project carries out mathematical studies of physical, biological and social dynamics systems in heterogeneous and random environments. Such systems are ubiquitous in nature, and include examples as diverse as ecology, economic growth, and fluid turbulence. 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, and numerical simulation of the microscopic details of the solutions is beyond reach even of the modern computers. A typical propagation distance may be of the order of hundreds or thousands of wavelengths and as many correlation lengths of random fluctuations. This necessitates the use of various approximate macroscopic effective models in practice. 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 for equations of the parabolic type, starting with the random heat equation, and then for nonlinear problems arising in the fluid dynamics and reaction-diffusion modeling, with the focus on very long time scales, when fluctuations in the solutions start building up in ways beyond the classical central limit theorem time scales. The second part of the project investigates the qualitative behavior of the solutions of systems of integro-differential equations that arise in macroeconomics. Such equations model phenomena from the diffusion of knowledge and GDP growth to the international trade. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.