Pseudospectral bound and transition threshold for the 3D Kolmogorov flow

In this paper, we establish the pseudospectral bound for the linearized operator of the Navier-Stokes equations around the 3D Kolmogorov flow. Using the pseudospectral bound and the wave operator method introduced in [LWZ], we prove the sharp enhanced dissipation rate for the linearized Navier-Stokes equations. As an application, we prove that if the initial velocity satisfies $\parallel U_0-(k_f^{-2}\sin (k_{f}y),0,0){\parallel }_{H^2}\le c{\nu }^{\frac{7}{4}}$ ($\nu$ the viscosity coefficient) and $k_f\in (0,1)$, then the solution does not transition away from the Kolmogorov flow.