  We study the three-dimensional Navier--Stokes equations of rotating
incompressible viscous fluids with periodic boundary conditions. The asymptotic
expansions, as time goes to infinity, are derived in all Gevrey spaces for any
Leray-Hopf weak solutions in terms of oscillating, exponentially decaying
functions. The results are established for all non-zero rotation speeds, and
for both cases with and without the zero spatial average of the solutions. Our
method makes use of the Poincar\'e waves to rewrite the equations, and then
implements the Gevrey norm techniques to deal with the resulting time-dependent
bi-linear form. Special solutions are also found which form infinite
dimensional invariant linear manifolds.

Analysis of PDEs