PDEs with highly oscillatory coefficients: homogenization and beyond

University Of Chicago, Chicago IL

Investigators

Abstract

This research is directed to development of new mathematical tools for investigation of important phenomena where the parameters determining these phenomena and the environment change rapidly in time and/or space. Such phenomena are ubiquitous, for example, in material science, atmospheric science, combustion, biomedical imaging, and the dynamics of financial markets. In this context the rapidity is determined by the scale: it might be days or hours for atmospheric phenomena, or seconds and milliseconds for chemical processes. In mathematical terms these phenomena are modeled by partial differential equations (PDE) with oscillatory coefficients. This research will foster both qualitative and quantitative understanding of solutions of underlying PDE through mathematical analysis and computer simulations. In particular, the project will result in the design of multi-scale numerical methods to simulate heterogeneous PDEs; performance of the methods in capturing the large-scale behavior and certain micro-scale information about the fluctuations of the solutions will be evaluated. The Principal Investigator will carry out the following studies: (i) develop homogenization theory for Hamilton-Jacobi equations in dynamic random environments where the Hamiltonian and/or the diffusion coefficients are highly oscillatory in space and time; in particular, new techniques will be developed to overcome the lack of uniform Lipschitz bounds due to the time dependence of the cell problem; (ii) for linear and semi-linear elliptic equations with random potential, for which effective potential is given by averaging, characterize the limiting probability distribution of the homogenization error, and generalize the studies to the case where the differential operator involves heterogeneous coefficients; (iii) study multi-scale numerical methods and evaluate their performance in capturing not only the macro-scale property of the heterogeneous PDEs but also certain micro-scale information about the fluctuations of the solutions, and to improve such methods.