Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on the two-dimensional torus

We consider the advection-diffusion equation on 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 |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.The paper uses a mixture of Classical Dynamical systems, Stochastic Dynamics, and PDE methods to mix a Lagrangian and an Eulerian perspective. If time permits I will also describe some even more recent work on Anomalous dissipation in a related setting. This is joint work with Tarek M. Elgindi and Kyle Liss from Duke.