Transition threshold for the 3D Couette flow in Sobolev space

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number $\text{Re}$. It was proved that if the initial velocity $v_0$ satisfies $\mathrm{\parallel }{v}_0-(y,0,0){\mathrm{\parallel }}_{H^2}\le c_0{\text{Re}}^{-1}$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture in physical literatures.