Stability threshold of the two-dimensional Couette flow in the whole plane

In this paper, we study the stability threshold for the two-dimensional Couette flow in the whole plane. Our main result establishes that the asymptotic stability threshold is at most $\frac{1}{3}+$ for Sobolev perturbations with additional control over low horizontal frequencies, aligning with the threshold results in periodic domains. As a secondary outcome of our approach, we also prove the asymptotic stability for perturbations in weak Sobolev regularity with size ${\nu }^{\frac{1}{2}}$.