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 $C^1$ perturbation. In particular, we extend well known finite-dimensional results to the setting...
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| Format: | Journal article |
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1998
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| _version_ | 1826260425991258112 |
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| author | Jones, D Shkoller, S |
| author_facet | Jones, D Shkoller, S |
| author_sort | Jones, D |
| collection | OXFORD |
| description | 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 $C^1$ 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 which possess them. te |
| first_indexed | 2024-03-06T19:05:28Z |
| format | Journal article |
| id | oxford-uuid:14f76dc2-5807-4ee4-8f77-36e5fb946808 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:05:28Z |
| publishDate | 1998 |
| record_format | dspace |
| spelling | oxford-uuid:14f76dc2-5807-4ee4-8f77-36e5fb9468082022-03-26T10:22:49ZPersistence of invariant manifolds for nonlinear PDEsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:14f76dc2-5807-4ee4-8f77-36e5fb946808Symplectic Elements at Oxford1998Jones, DShkoller, SWe 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 $C^1$ 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 which possess them. te |
| spellingShingle | Jones, D Shkoller, S Persistence of invariant manifolds for nonlinear PDEs |
| title | Persistence of invariant manifolds for nonlinear PDEs |
| title_full | Persistence of invariant manifolds for nonlinear PDEs |
| title_fullStr | Persistence of invariant manifolds for nonlinear PDEs |
| title_full_unstemmed | Persistence of invariant manifolds for nonlinear PDEs |
| title_short | Persistence of invariant manifolds for nonlinear PDEs |
| title_sort | persistence of invariant manifolds for nonlinear pdes |
| work_keys_str_mv | AT jonesd persistenceofinvariantmanifoldsfornonlinearpdes AT shkollers persistenceofinvariantmanifoldsfornonlinearpdes |