Exponential mixing for random dynamical systems and an example of Pierrehumbert

TL;DR

This paper develops a robust framework for exponential mixing in random dynamical systems on compact manifolds and applies it to a classical example, solving a longstanding open problem.

Contribution

It introduces a dynamics-based framework for constructing universal exponential mixers and applies it to Pierrehumbert's example, establishing exponential mixing for smooth incompressible flows.

Findings

Established exponential mixing for Pierrehumbert's flow

Provided a general toolbox for constructing smooth universal mixers

Solved a longstanding open problem in fluid dynamics

Abstract

We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain while also providing a toolbox for constructing such smooth universal mixers in all dimensions.