  We prove that under certain stability and smoothing properties of the
semi-groups generated by the partial differential equations that we consider,
manifolds left invariant by these flows persist under $C^1$ perturbation. In
particular, we extend well known finite-dimensional results to the setting of
an infinite-dimensional Hilbert manifold with a semi-group that leaves a
submanifold invariant. We then study the persistence of global unstable
manifolds of hyperbolic fixed-points, and as an application consider the
two-dimensional Navier-Stokes equation under a fully discrete approximation.
Finally, we apply our theory to the persistence of inertial manifolds for those
PDEs which possess them. te
