  The Navier-Stokes equations for viscous, incompressible fluids are studied in
the three-dimensional periodic domains, with the body force having an
asymptotic expansion, when time goes to infinity, in terms of power-decaying
functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to
have an asymptotic expansion of the same type in the same space, which is
uniquely determined by the force, and independent of the individual solutions.
In case the expansion is convergent, we show that the next asymptotic
approximation for the solution must be an exponential decay. Furthermore, the
convergence of the expansion and the range of its coefficients, as the force
varies, are investigated.
