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 |log υ|, where υ is the diffusivity parameter. This is the optimal decay rate as υ → 0 for uniformly-in-time Lipschitz velocity fields. We also establish exponential mixing for the υ = 0 problem.