Asymptotic expansions in time for rotating incompressible viscous fluids

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 terms oscillating, exponentially decaying functions. results established non-zero rotation speeds, and both cases without zero spatial average solutions. Our method makes use Poincar\'e waves rewrite equations, then implements norm techniques deal resulting time-dependent bi-linear form. Special also found which form infinite dimensional invariant linear manifolds.