  We study the three-dimensional Navier-Stokes equations in a periodic domain
with the force decaying in time. Although the force has a certain coherent
decay, as time tends to infinity, it can be too complicated for the previous
theory of asymptotic expansions to be applicable. To deal with this issue, we
systematically develop a new theory of asymptotic expansions containing the
so-called subordinate variables which can be defined recursively. We apply it
to obtain an asymptotic expansion for any Leray-Hopf weak solutions. The
expansion, in fact, is constructed explicitly and the impact of the subordinate
variables can be clearly specified. The complexifications of the Gevrey-Sobolev
spaces, and of the Stokes and bilinear operators of the Navier-Stokes equations
are utilized to facilitate such a construction.
