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 we obtain that the Couette flow $(y,0)$ is stable under the space of ${\mathcal{D}}^{1,q}({\mathbb{R}}^2)$ for any $1<q<\mathrm{\infty }$ and unstable in the space of ${\mathcal{D}}^{1,\mathrm{\infty }}({\mathbb{R}}^2)$. A key observation is the anisotropic cut-off function. We also consider the Poiseuille flow $(y^2,0)$, which is stable in ${\mathcal{D}}^{1,q}({\mathbb{R}}^2)$ with $\frac{4}{3}<q\le 4.$