We prove that under certain stability and smoothing properties of the semi-
groups generated by the partial differential equations that we consider, ma
nifolds left invariant by these flows persist under C-1 perturbation, In pa
rticular, we extend well-known finite-dimensional results to the setting of
an infinite-dimensional Hilbert manifold with a semi-group that leaves a s
ubmanifold invariant. We then study the persistence of global unstable mani
folds of hyperbolic fixed points, and as an application consider the two-di
mensional Navier-Stokes equation under a fully discrete approximation. Fina
lly, we apply our theory to the persistence of inertial manifolds for those
PDEs that possess them.