Uniform vorticity depletion and inviscid damping for periodic shear flows in the high Reynolds number regime

We study the dynamics of the two dimensional Navier-Stokes equations linearized around a shear flow on a (non-square) torus which possesses exactly two non-degenerate critical points. We obtain linear inviscid damping and vorticity depletion estimates for the linearized flow that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The main task is to understand the associated Rayleigh and Orr-Sommerfeld equations, under the natural assumption that the linearized operator around the shear flow in the inviscid case has no discrete eigenvalues. The key difficulty is to understand the behavior of the solution to Orr-Sommerfeld equations in three distinct regimes depending on the spectral parameter: the non-degenerate case when the spectral parameter is away from the critical values, the intermediate case when the spectral parameter is close to but still separated from the critical values, and the most singular case when the spectral parameter is inside the viscous layer.