Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on T2

We consider the advection-diffusion equation on T2 with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale |logν|, where ν is the diffusivity parameter. This is the optimal decay rate as ν→0 for uniformly-in-time Lipschitz velocity fields. We also establish exponential mixing for the ν=0 problem.