Persistence of Invariant Manifolds for Nonlinear PDEs

Source abstract

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 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 that possess them.