Quantitative stability for the 2D Couette flow on the infinite channel with non‐slip boundary condition

Abstract In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel with non‐slip boundary condition. Compared to the case , we establish the stability in the context of long wave associated with the frequency range by developing the resolvent estimate argument. The new ingredient is to discover the key division point at in the frequency interval (0,1) by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval (0,1), and then we establish the space–time estimates on the low‐frequency and the intermediate frequency , respectively. As an application of the space–time estimates, we obtain the non‐linear transition threshold to be . Meanwhile, we also show that when the frequencies , the enhanced dissipation effect occurs for the linearized Navier–Stokes equations.