The study of 2d shear flows in the context of ideal fluid or viscous fluid in the high Reynolds number regime is a classical topic in the area of hydrodynamic stability. The early pioneering works focused primarily (but not exclusively) on the possible appearances of unstable eigenvalues. For spectrally stable shear flows, it is reasonable to expect certain stability, at least on the linearized level. A fundamental stabilizing mechanism is the inviscid damping for the Euler equation, i.e., energy transfer to high frequencies on the vorticity level leading to decay of velocity fields. Moreover, for slightly viscous flow and flows with critical points additional important physical phenomena appear, such as vorticity depletion and enhanced dissipation, in addition to the persistence of inviscid damping. In this talk, we review a uniform framework for addressing these issues and discuss some remaining open problems.

Part of the talk is based on joint works with Alex Ionescu, and Sameer Iyer.