Topics in Analysis

In this three-part lecture series, we will present a series of works by Bedrossian, Blumenthal and Punshon-Smith on the chaotic mixing and enhanced dissipation properties of a passive tracer subject to the motion of an ergodic Markovian flow of spatially regular incompressible velocity fields on a compact domain. Such flows crucially include the motion of a velocity field solving the stochastic Navier-Stokes equations in $T^{2}$ (or $T^{3}$ with hyperviscosity), but can be more general or much simpler in character. We will show how tools from the smooth ergodic theory of random dynamical systems can be developed to show that these flows "generically'' give rise to ergodic flows of volume preserving diffeomorphisms (Lagrangian flow) that have a positive Lyapunov exponent and are almost surely exponentially mixing. Moreover, when diffusion is added, they exhibit optimal enhanced $L^{2}$ dissipation, whereby the interaction between advection and diffusion causes a passive tracer to decay on much faster time scales than the heat equation. This enhancement time scale is optimal in the class of Lipschitz velocity fields and provides the first examples of flows with such properties beyond the uniformly hyperbolic setting.

In Lecture 3, we will describe how, for such flows, one can turn the weaker local information of a positive Lyapunov exponent into the stronger global information about mixing and exponential correlation decay using powerful tools from the ergodic theory of Markov chains. In particular, we will develop techniques that use the Harris theorem and large deviation theory to turn a positive Lyapunov exponent into a spectral gap for the Markov process of two point particle motion, which is a sufficient condition for almost sure exponential mixing. Time permitting, we will explain how to prove the remarkable fact that this spectral gap is stable under the addition of diffusion (a singular perturbation) and crucially gives rise to an exponential mixing rate that is uniform in the strength of the diffusivity. This very strong uniform-in-diffusivity mixing property can be seen to easily imply an optimal $L^{2}$ enhanced dissipation effect and is also the crucial ingredient in the proof of the emergence of the -1 law in the power spectrum of passive scalar turbulence.