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In this paper, we establish the pseudospectral bound for the linearizedoperator of the Navier-Stokes equations around the 3D Kolmogorov flow. Usingthe pseudospectral bound and the wave operator method introduced in [LWZ], weprove the sharp enhanced dissipation rate for the linearized Navier-Stokesequations. As an application, we prove that if the initial velocity satisfies$\big\| U_0-\big(k_f^{-2}\sin(k_fy),0,0\big)\big\|_{H^2}\le c\nu^{\frac{7}{4}}$($\nu$ the viscosity coefficient) and $k_f\in (0,1)$, then the solution doesnot transition away from the Kolmogorov flow.

Pseudospectral bound and transition threshold for the 3D Kolmogorov flow