Enhanced dissipation for time-periodic alternating shear flows
We consider the long-time behavior of solutions to the advection-diffusion with an incompressible velocity field. An intriguing feature of this equation is that the formation of small scales due to advection can cause solutions to decay much faster than on the standard diffusive timescale in the regime of weak diffusion. This phenomenon is often referred to as enhanced dissipation. In this talk, I will discuss recent joint work with Tarek Elgindi and Jonathan Mattingly in which we study enhanced dissipation on the periodic box for time-periodic, alternating piecewise linear shear flows. Our main result is an estimate on the convergence rate of solutions to equilibrium which is optimal with respect to scaling in the small diffusive parameter. The proof is based on the probabilistic representation formula of the advection-diffusion equation and ideas from dynamical systems.