Persistence of invariant manifolds for nonlinear PDEs

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.