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In this paper, we study the transition threshold of the 3D Couette flow inSobolev space at high Reynolds number $\text{Re}$. It was proved that if theinitial 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 doesnot transition away from the Couette flow. This result confirms the transitionthreshold conjecture in physical literatures.

Transition threshold for the 3D Couette flow in Sobolev space