In this talk we discuss the long time dynamics of linearized Navier Stokes equations near spectrally stable shear flows on a torus. Physically, there are two significant phenomena: inviscid damping of stream and velocity fields over time and depletion of vorticity from critical points of the shear flow. In recent years the mechanisms for both phenomena have been gradually understood in the context of ideal fluid, e.g. when the viscosity of the fluid is assumed to be zero. A natural question is whether the same phenomena would occur in the viscous case when the viscosity is small but nonzero. We will discuss some recent progress in this problem, and present an upcoming result which establishes inviscid damping, vorticity depletion and enhanced dissipation bounds that are uniform with respect to viscosity. Joint work with Rajendra Beekie and Shan Chen.
Department of Mathematics