Uniform-in-diffusivity mixing by shear flows: stochastic and dynamical perspectives

TL;DR

This paper investigates the mixing of passive scalars by shear flows with weak diffusion, providing new proofs and insights into the uniform-in-diffusivity mixing rates using stochastic and dynamical systems methods.

Contribution

It offers two novel proofs of uniform-in-diffusivity mixing rates for shear flows, one using stochastic integration-by-parts and the other from a dynamical systems perspective.

Findings

Recovered sharp mixing rate for shear flows with finitely many critical points

Provided a proof of shear-induced mixing in the zero-diffusivity case

Answered an open question regarding optimal regularity assumptions

Abstract

We study passive scalar mixing by parallel shear flows in the presence of weak molecular diffusion. We recover the sharp uniform-in-diffusivity mixing rate for shear flows with finitely many critical points, recently proven in [1]. Our approach is based on the stochastic representation formula of the associated advection-diffusion equation and yields two short proofs. The first uses a stochastic integration-by-parts argument and gives optimal mixing under the weakest regularity assumption required in the zero-diffusion case, answering Question II in [1, Section 4]. The second adopts a dynamical systems perspective and provides a proof of shear-induced mixing that, to our knowledge, is new even in the zero-diffusivity setting.