Jones, D. A. and Stuart, A. M. (1995) Attractive Invariant Manifolds under Approximation. Inertial Manifolds. Journal of Differential Equations, 123 (2). pp. 588-637. ISSN 0022-0396. doi:10.1006/jdeq.1995.1174. https://resolver.caltech.edu/CaltechAUTHORS:20170612-064633625
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Abstract
A class of nonlinear dissipative partial differential equations that possess finite dimensional attractive invariant manifolds is considered. An existence and perturbation theory is developed which unifies the cases of unstable manifolds and inertial manifolds into a single framework. It is shown that certain approximations of these equations, such as those arising from spectral or finite element methods in space, one-step time-discretization or a combination of both. also have attractive invariant manifolds. Convergence of the approximate manifolds to the true manifolds is established as the approximation is refined. In this part of the paper applications to the behavior of inertial manifolds under approximation are considered. From this analysis deductions about the structure of the attractor and the flow on the attractor under discretization can be made.
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| Additional Information: | © 1995 Academic Press. Received January 18. 1994; revised June 3, 1994. Part of this work was completed while D.A.J. enjoyed the hospitality of the Center for Turbulence Research at Stanford University. D.A.J. was partially supported by the Department of Energy "Computer Hardware, Advanced Mathematics, Model Physics" (CHAMMP) research program as part of the U.S. Global Change Research Program. The work of A.M.S. is supported by the Office of Naval Research. Contract N00014-92-J-l876 and by the National Science Foundation. Contract DMS-9201727. The authors are also grateful to John Toland for the helpful discussions. | |||||||||
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| Issue or Number: | 2 | |||||||||
| DOI: | 10.1006/jdeq.1995.1174 | |||||||||
| Record Number: | CaltechAUTHORS:20170612-064633625 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20170612-064633625 | |||||||||
| Official Citation: | D.A. Jones, A.M. Stuart, Attractive Invariant Manifolds under Approximation. Inertial Manifolds, Journal of Differential Equations, Volume 123, Issue 2, December 1995, Pages 588-637, ISSN 0022-0396, https://doi.org/10.1006/jdeq.1995.1174. (http://www.sciencedirect.com/science/article/pii/S0022039685711746) | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 78078 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | Ruth Sustaita | |||||||||
| Deposited On: | 12 Jun 2017 16:22 | |||||||||
| Last Modified: | 15 Nov 2021 17:36 |
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