  We study the long-time behavior of spatially periodic solutions of the
Navier-Stokes equations in the three-dimensional space. The body force is
assumed to possess an asymptotic expansion or, resp., finite asymptotic
approximation, in either Sobolev or Gevrey spaces, as time tends to infinity,
in terms of polynomial and decaying exponential functions of time. We establish
an asymptotic expansion, or resp., finite asymptotic approximation, of the same
type for the Leray-Hopf weak solutions. This extends the previous results,
obtained in the case of potential forces, to the non-potential force case,
where the body force may have different levels of regularity and asymptotic
approximation. In fact, our analysis identifies precisely how the structure of
the force influences the asymptotic behavior of the solutions.
