Persistence of invariant manifolds for nonlinear PDEs

1998/07/17 by Jones, Don A., Shkoller, Steve
#35K50 #58B99 #76D05 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.math/9807090

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 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 which possess them. te

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