  Spectral subspaces of a linear dynamical system identify a large class of
invariant structures that highlight/isolate the dynamics associated to select
subsets of the spectrum. The corresponding notion for nonlinear systems is that
of spectral submanifolds -- manifolds invariant under the full nonlinear
dynamics that are determined by their tangency to spectral subspaces of the
linearized system. In light of the recently-emerged interest in their use as
tools in model reduction, we propose an extension of the relevant theory to the
realm of fluid dynamics. We show the existence of a large (and the most
pertinent) subclass of spectral submanifolds and foliations - describing the
behaviour of nearby trajectories - about fixed points and periodic orbits of
the Navier-Stokes equations. Their uniqueness and smoothness properties are
discussed in detail, due to their significance from the perspective of model
reduction. The machinery is then put to work via a numerical algorithm
developed along the lines of the parameterization method, that computes the
desired manifolds as power series expansions. Results are shown within the
context of 2D channel flows.
