Diffusion and Mixing in Fluid Flow: A Review

Abstract

This note is a review of a series of results on the interaction between diffusion and fluid flow that have been presented by the author at the International Congress in Mathematical Physics in Rio, 2006. The main object of study is the enhancement of diffusive mixing by a fast incompressible flow. Due to its physical relevance, the subject has been studied in detail from different angles. Here, we describe some of the recent work which combines PDE, functional analysis and dynamical systems theory by trying to establish links between diffusion enhancement and mixing properties inherent to the dynamical system generated by the flow. The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form Γ+iAL with a negative unbounded self-adjoint operator Γ, a self-adjoint operator L, and parameter A≫1. In particular, they employ the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the Hamiltonian (related to a classical theorem of Wiener on Fourier transforms of measures).