Optimal mixing and enhanced dissipation for a smooth velocity field in any spatial dimension

Abstract

In this note, we consider the advection-diffusion equation on the $d$-dimensional torus, where $d$ is arbitrary. First, for the associated transport equation, we provide an explicit example of a space-time smooth, divergence-free, random velocity field that is a universal exponential mixer. Second, we set the molecular diffusivity to a (small) positive value, and we show that the same velocity field is dissipation enhancing with optimal rate. Our construction is based on the Random Dynamical Systems viewpoint, and relies on the uniform ergodicity of the two-point Markov process associated with the two-dimensional Pierrehumbert model.