Anomalous diffusion in strong cellular flows: averaging and homogenization

Abstract

Title of dissertation: ANOMALOUS DIFFUSION IN STRONG CELLULAR FLOWS: AVERAGING AND HOMOGENIZATION Zsolt Pajor-Gyulai, Doctor of Philosophy, 2015 Dissertation directed by: Professor Leonid Koralov Department of Mathematics This thesis considers the possible limit behaviors of a strong Hamiltonian cellular flow that is subjected to a Brownian stochastic perturbation. Three possible limits are identified. When long timescales are considered, the limit behavior is described by classical homogenization theory. In the intermediate (finite) time case, it is shown that the limit behavior is anomalously diffusive. This means that the limit is given by a Brownian motion that is time changed by the local time of a process on the graph which is associated with the structure of the unperturbed flow lines (Reeb graph) that one obtains by Freidlin-Wentzell type averaging. Finally, we consider the case when the motion starts close to, or on, the cell boundary and derive a limit for the displacement on timescales of order e where α ∈ (0, 1) (modulo a logarithmic correction to compensate for the slowdown of the flow near the saddle points of the Hamiltonian). The latter two cases are novel results obtained by the author and his collaborators ( [1]). We also consider two applications of the above results to associated partial differential equation (PDE) problems. Namely, we study a two-parameter averaginghomogenization type elliptic boundary value problem and obtain a precise description of the limit behavior of the solution as a function of the parameters using the well-known stochastic representation. Additionally, we study a similar parabolic Cauchy problem. ANOMALOUS DIFFUSION IN STRONG CELLULAR FLOWS: AVERAGING AND HOMOGENIZATION by Zsolt Pajor-Gyulai Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2015 Advisory Committee: Professor Leonid Koralov, Chair/Advisor Professor Sandra Cerrai Professor Mark Freidlin Professor Konstantina Trivisa Professor Ilya Ryzhov c © Copyright by Zsolt Pajor-Gyulai 2015