Global stability of large Fourier mode for 3-D Navier-Stokes equations in cylindrical coordinates

We prove the global existence of strong solutions to 3-D incompressible Navier-Stokes equations in the whole space or in the cylindrical domain { x = (x1,x2,x3) ∈ ℝ3: r=\sqrt{x_1^2+\ x_2^2}\ <1, x3 ∈ ℝ }, provided that the initial data are of the form: A(r,z) cos Nθ + B(r,z) sin Nθ, with arbitrary axially symmetric functions A(r,z) and B(r,z), but N needs to be large enough. Moreover, the corresponding solution has almost the same frequency N for any positive time. Precisely, the kN-th Fourier coefficients decay exponentially in k. The similar result also holds for 3-D anisotropic Navier-Stokes equations with diffusion only in x1 and x2 variables.