Asymptotic stability of two-dimensional Couette flow in a viscous fluid

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's generally known 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 size and the regularity of the initial perturbation that makes the nonlinear asymptotic stability holds. More precisely, we proved that if the initial perturbation is in some Gevrey-$\frac{1}{s}$ class with size $\epsilon\nu^{\beta}$ where $s\geq \frac{1-3\beta}{2-3\beta}$ and $\beta\in [0,\frac{1}{3}]$, then the nonlinear asymptotic stability holds.