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 $\|v_0-(y,0,0)\|_{ H^2}\le c_0\text{ Re}^{ -1}$, 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.
Categories: math.AP