In this paper, we study the nonlinear asymptotic stability of Couette flow for the two-dimensional Navier-Stokes equation with small viscosity \(\nu >0\) in \(\mathbb {T}\times \mathbb {R}\) . It is well known that the nonlinear asymptotic stability of the Couette flow depends closely on the size and regularity of the initial perturbation, which yields the stability threshold problem. This work studies the relationship between the regularity and the size of the initial perturbation that makes the nonlinear asymptotic stability hold. More precisely, we prove that if the initial perturbation is in some Gevrey- \(\frac{1}{s}\) class with size \(\varepsilon \nu ^{\beta }\) where \(s\in [0,\frac{1}{2}]\) and \(\beta \ge \frac{1-2s}{3-3s}\) , then the nonlinear asymptotic stability holds. We think this index is sharp.
