  We study the long-time dynamics of the Navier-Stokes equations in the
three-dimensional periodic domains with a body force decaying in time. We
introduce appropriate systems of decaying functions and corresponding
asymptotic expansions in those systems. We prove that if the force has a
large-time asymptotic expansion in Gevrey-Sobolev spaces in such a general
system, then any Leray-Hopf weak solution admits an asymptotic expansion of the
same type. This expansion is uniquely determined by the force, and independent
of the solutions. Various applications of the abstract results are provided
which particularly include the previously obtained expansions for the solutions
in case of power decay, as well as the new expansions in case of the
logarithmic and iterated logarithmic decay.
