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, manifolds left invariant by these flows persist under C¹ 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 that possess them.