Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$

TL;DR

This paper proves optimal enhanced dissipation and exponential mixing for a time-periodic Lipschitz velocity field on the torus, with decay rates matching the best possible as diffusivity vanishes.

Contribution

It establishes the optimal decay rate for enhanced dissipation in a Lipschitz, time-periodic setting, extending understanding of mixing and dissipation in such flows.

Findings

Enhanced dissipation occurs on the $| ext{log} u|$ timescale.

Exponential mixing is proven for the zero-diffusivity case.

Optimal decay rate is achieved as diffusivity tends to zero.

Abstract

We consider the advection-diffusion equation on $\mathbb{T}^2$ with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale $|\log \nu|$, where $\nu$ is the diffusivity parameter. This is the optimal decay rate as $\nu \to 0$ for uniformly-in-time Lipschitz velocity fields. We also establish exponential mixing for the $\nu = 0$ problem.