Stability of the Couette flow under the 2D steady Navier-Stokes flow

In this note, we investigate the stability property of shear flows under the 2D stationary Navier-Stokes equations, and obtain that the Couette flow (y, 0) is stable under the space of D-1,D-q(R-2) for any 1<q<infinity and unstable in the space of D-1,D-infinity(R-2), which is sharp in this sense. A key observation is the choice of the anisotropic cut-off function. The Poiseuille flow (y(2), 0) is also considered as a by-product, which is stable in the space of D-1,D-q(R-2) with 4/3<q <= 4 via a lemma of Fefferman-Stein.