  First, we consider Kolmogorov flow (a shear flow with a sinusoidal velocity
profile) for 2D Navier-Stokes equation on a torus. Such flows, also called bar
states, have been numerically observed as one type of metastable states in the
study of 2D turbulence. For both rectangular and square tori, we prove that the
non-shear part of perturbations near Kolmogorov flow decays in a time scale
much shorter than the viscous time scale. The results are obtained for both the
linearized NS equations with any initial vorticity in L^2, and the nonlinear NS
equation with initial L^2 norm of vorticity of the size of viscosity. In the
proof, we use the Hamiltonian structure of the linearized Euler equation and
RAGE theorem to control the low frequency part of the perturbation. Second, we
consider two classes of shear flows for which a sharp stability criterion is
known. We show the inviscid damping in a time average sense for non-shear
perturbations with initial vorticity in L^2. For the unstable case, the
inviscid damping is proved on the center space. Our proof again uses the
Hamiltonian structure of the linearized Euler equation and an instability index
theory recently developed by Lin and Zeng for Hamiltonian PDEs.
