Sharp uniform-in-diffusivity mixing rates for passive scalars in parallel shear flows

TL;DR

This paper establishes optimal uniform-in-diffusivity mixing rates for passive scalars in shear flows, revealing how the mixing rate depends on the shear profile's degeneracy and providing detailed asymptotic descriptions.

Contribution

It proves the optimal mixing rate bounds for passive scalars in shear flows and offers a resolvent-based analysis along with asymptotic descriptions for solutions.

Findings

Optimal mixing rate $ orm{f}_{H^{-1}} o 0$ as $t o abla$

Dependence of mixing rate on shear profile degeneracy $N$

Asymptotic behavior characterized by shear layers around critical points

Abstract

We consider the advection-diffusion equation describing the evolution of a passive scalar in a background shear flow. We prove the optimal uniform-in-diffusivity mixing rate $\| f \|_{H^{-1}} \lesssim \langle t \rangle^{-1/(N+1)}$, $t \geq 0$, where $N$ is the maximal order of vanishing of the derivative $b'(y)$ of the shear profile, e.g., $N=1$ for plane Pouseille flow. Our proof is based on the description of the solution in terms of resolvents and involves pointwise estimates on the resolvent kernel. In the non-degenerate case, we further give a rigorous asymptotic description of generic solutions in terms of shear layers localized around the critical points. This verifies formal asymptotics in [McLaughlin-Camassa-Viotti, \textit{Physics of Fluids}, 22(11), 2010].