Transition threshold for the 2-D Couette flow in whole space via Green's function

In this paper, we investigate the transition threshold problem concerning the 2-D Navier-Stokes equations in the context of Couette flow $(y,0)$ at high Reynolds number $Re$ in whole space. By utilizing Green’s function estimates for the linearized equations around Couette flow, we initially establish refined dissipation estimates for the linearized Navier-Stokes equations with a precise decay rate $(1+t{)}^{-1}\mathrm{.}$ As an application, we prove that if the initial perturbation of vorticity satisfies$$\mathrm{\parallel }{\omega }_0{\mathrm{\parallel }}_{H^1\cap L^1}\le c_0{\nu }^{\frac{3}{4}}$$ for some small constant $c_0$ independent of the viscosity $\nu$, then we can reach the conclusion that the solution remains within $O\left({\nu }^{\frac{3}{4}}\right)$ of the Couette flow.