Exponential Lower Bounds for the Advection-Diffusion Equation with Shear Flows

TL;DR

This paper proves that in a 2D advection-diffusion system with shear flows, the $L^2$ norm of solutions cannot decay faster than exponentially, indicating limits on how quickly mixing can occur due to diffusion.

Contribution

It establishes the first exponential lower bounds for the decay of solutions in the presence of shear flows, highlighting fundamental limits of passive scalar mixing.

Findings

Solutions' $L^2$ norm decays at most exponentially over time.

Diffusion cannot accelerate mixing beyond exponential rates.

Passive scalar mixing is fundamentally limited by shear flows.

Abstract

In this paper, we prove that the $L^2$ norm of spatial mean-free solutions to the advection--diffusion equation on $\mathbb{T}^2$ with shear drifts satisfies an \emph{exponential lower bound} in time. This lower bound shows that diffusion can fundamentally suppress passive-scalar mixing.