Stability and instability of the Couette flow in the Navier-Stokes equations

The plane, periodic Couette flow has served as a canonical problem in the field of hydrodynamic stability since the late 19th century. Obtaining a precise understanding of the stability and instability of this flow in the nonlinear equations has however remained elusive in both 2D and 3D. In this talk I will discuss some of the recent progress we have made in this direction in both 2D (joint with Nader Masmoudi, Vlad Vicol and Fei Wang) and 3D (joint with collaborators Pierre Germain and Nader Masmoudi). The dynamics of solutions have been determined in a variety of settings, and are governed by several important effects: namely inviscid damping and mixing-enhanced dissipation. Comparisons and contrasts will be made with recent results on Landau damping in kinetic theory.