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

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 $\mathrm{\mid }\log \nu \mathrm{\mid }$, 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.