Persistence of invariant manifolds for nonlinear PDEs

Donald Jones, Steve Shkoller

Research output: Contribution to journalArticle

5 Scopus citations

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 that possess them.

Original languageEnglish (US)
Pages (from-to)27-67
Number of pages41
JournalStudies in Applied Mathematics
Volume102
Issue number1
DOIs
StatePublished - Jan 1999

ASJC Scopus subject areas

  • Applied Mathematics

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