  In this paper, we first prove the global existence of strong solutions to 3-D
incompressible Navier-Stokes equations with solenoidal initial data, which
writes in the cylindrical coordinates is of the form: $A(r,z)\cos N\theta
+B(r,z)\sin N\theta,$ provided that $N$ is large enough. In particular, we
prove that the corresponding solution has almost the same frequency $N$ for any
positive time. The main idea of the proof is first to write the solution in
trigonometrical series in $\theta$ variable and estimate the coefficients
separately in some scale-invariant spaces, then we handle a sort of weighted
sum of these norms of the coefficients in order to close the a priori estimate
of the solution. Furthermore, we shall extend the above well-posedness result
for initial data which is a linear combination of axisymmetric data without
swirl and infinitely many large mode trigonometric series in the angular
variable.
