Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$C^{1}$ approximations of inertial manifolds
via finite differences

Author: Kazuo Kobayasi
Journal: Proc. Amer. Math. Soc. 127 (1999), 1143-1150
MSC (1991): Primary 47H20; Secondary 35K55
MathSciNet review: 1610800
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct an inertial manifold for the evolution equation as a limit of the inertial manifolds for the difference approximations of the Trotter-Kato type and show that this limit is taken in a $C^{1}$ topology.

References [Enhancements On Off] (What's this?)

  • 1. S.N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J.Differential Equations 74 (1988), 285-317. MR 89h:58163
  • 2. S.N. Chow, K. Lu and G.R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl. 169 (1992), 283-321. MR 94a:58172
  • 3. F. Demengel and J.M. Ghidaglia, Inertial manifolds for partial differential equations under time-discretization: Existence and applications, J. Math. Anal. Appl. 155 (1991), 177-225. MR 92e:35079
  • 4. C. Foias, M.S. Jolly, I.G. Kevrekidis and E.S. Titi, Dissipativity of numerical schemes, Nonlinearity 4 (1991), 591-613. MR 92f:65099
  • 5. C. Foias, G. Sell and R. Temam, Inertial manifolds for nonlinear evolutional equations, J. Differential Equations 73 (1988), 309-353.
  • 6. C. Foias and E.S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity 4 (1991), 135-153. MR 92a:65241
  • 7. E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, RI, 1957. MR 54:11077; MR 19:664d
  • 8. M.S. Hirsch, C.C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics Vol.583, Springer-Verlag, New York, 1977. MR 58:18595
  • 9. D.A. Jones and A.M. Stuart, Attractive invariant manifolds under approximation. Inertial manifolds, J. Differential Equations 123 (1995), 588-637. MR 96j:34110
  • 10. D.A. Jones and E.S. Titi, $C^{1}$ approximation of inertial manifolds for dissipative nonlinear equations, J. Differential Equations 127 (1996), 54-86. MR 97g:58154
  • 11. K. Kobayasi, Convergence and approximation of inertial manifolds for evolution equations, Differential and Integral Equations 8 (1995), 1117-1134. MR 96c:34131
  • 12. K. Kobayasi, Inertial manifolds for discrete approximations of evolution equations: Convergence and approximations, Advances in Math. Sci. Appl. 3 (1993/1994), 161-189. MR 95d:34107
  • 13. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. MR 85g:47061
  • 14. V.A. Pliss and G.R. Sell, Perturbations of attractors of differential equations, J. Differential Equations 92 (1991), 100-124. MR 92f:58094

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H20, 35K55

Retrieve articles in all journals with MSC (1991): 47H20, 35K55

Additional Information

Kazuo Kobayasi
Affiliation: Department of Mathematics, School of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-Ku, Tokyo 169-8050, Japan

Keywords: Inertial manifold, long-time behavior, finite dynamical system, evolution equation
Received by editor(s): July 29, 1997
Additional Notes: This research was partially supported by Waseda University Grant for special Research Projects 97A-81.
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society