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

Abstract

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  1,𝑞 (ℝ 2 ) for any 1 < 𝑞 < ∞ and unstable in the space of  1,∞ (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 byproduct, which is stable in the space of  1,𝑞 (ℝ 2 ) with 4 3 < 𝑞 ≤ 4 via a lemma of Fefferman-Stein.