Description: Homogenization is a technique which is used to approximate macroscopic behaviour in the presence of micro scale fluctuations. A fundamental application to fluid dynamics was conceived by G. I. Taylor (JFM '53) by estimating the effective diffusivity of a solute diffusing in the presence of a laminar flow in a pipe. Unfortunately the length scales involved in typical oil pipelines are not long enough for this result to apply! This course will focus on few results that address regimes where the homogenization results don't apply.
Specifically, in the context of cellular flows, Young '88 observed both numerically and experimentally a robust and stable anomalous diffusive behaviour at time scales shorter than those required by homogenization results. In joint work with A. Novikov we prove that the variance of tracer particles on intermediate time scales is consistent with Young's observation. I will conclude with a description of recent work by Hairer, Koralov and Pajor-Gulai showing that the effective behaviour is a subordinated Brownian motion.