  We enable a theory of intrinsic asymptotic expansions for the steady state
solutions of the full Navier--Stokes equations. Such a theory was first
developed in \cite{FHJ} for Galerkin approximations. To overcome the lack of
local compactness in infinite dimensional spaces, we introduce the notion of
asymptotic expansions in nested spaces. When the inclusion maps between these
spaces are compact, we establish the existence of such an asymptotic expansion
for a subsequence of any bounded sequence. This consequently yields an
intrinsic asymptotic expansion in a single normed space. We apply this result
to the steady states of the Navier--Stokes equations by utilizing the spectral
fractional Sobolev spaces. In the case of 2D periodic boundary conditions, more
properties relating the terms of the asymptotic expansion are obtained.
