Discrete inertial manifolds

Mathematische Nachrichten

Volumn 281, Issue 6, 2008, Pages 847-878

  Pötzsche, Christian ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Discrete Ginzburg Landau equation; Discretization; Inertial manifold; Invariant fiber bundles; Normal hyperbolicity; Pullback attraction

Indexed keywords

EID: 55549083568     PISSN: 0025584X     EISSN: 15222616     Source Type: Journal    
DOI: 10.1002/mana.200710645     Document Type: Article

References (32)

1. R. H. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications (Springer, Berlin, 1988).
2. H. Amann, Ordinary Differential Equations: An Introduction to Nonlinear Analysis (de Gruyter, Berlin, 1990).
3. B. Aulbach, The fundamental existence theorem on invariant fiber bundles, J. Difference Equ. Appl. 3, 501-537 (1998).
4. B. Aulbach and T. Wanner, Invariant foliations and decoupling of non-autonomous difference equations, J. Difference Equ. Appl. 9, No. 5, 459-472 (2003).
5. W.-J. Beyn and B. M. Garay, Estimates of variable stepsize Runge-Kutta methods for sectorial evolution equations with nonsmooth data, Appl. Numer. Math. 41, No. 3, 369-400 (2002).
6. V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics (Amer. Math. Soc., Providence, RI, 2001).
7. 1 semigroups in Banach spaces, J. Differential Equations 139, 283-318 (1997).
8. J. B. Conway, A Course in Functional Analysis, 2nd edition (Springer, New York, 1990).
9. F. Demengel and J.-M. Ghidaglia, Intertial manifolds for partial differential evolution equations under time-discretization: Existence, convergence, and applications, J. Math. Anal. Appl. 155, 177-225 (1991).
10. J. L. M. van Dorsselaer and C. Lubich, Inertial manifolds of parabolic differential equations under higher-order discretizations, IMA J. Numer. Analysis 19, 455-471 (1999).
11. J. Duan, H. V. Ly, and E. S. Titi, The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation, Z. Angew. Math. Phys. 47, No. 3, 432-455 (1996).
12. A. Eden, B. Michaux, and J. M. Rakotoson, Semi-discretized nonlinear evolution equations as discrete dynamical systems and error analysis, Indiana Univ. Math. J. 39, No. 3, 737-783 (1990).
13. G. Farkas, Small delay inertial manifolds under numerics: A numerical structural stability result, J. Dynam. Differential Equations 14, No. 3, 549-588 (2002).
14. N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21, 193-226 (1971).
15. C. Foias, M. S. Jolly, I. Kevrekidis, and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity 4, No. 3, 591-613 (1991).
16. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics Vol. 840 (Springer, Berlin, 1981).
17. G. Iooss, Bifurcation of Maps and Applications (North-Holland, Amsterdam, 1979).
18. D. A. Jones, A. W. Stuart, and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, J. Math. Anal. Appl. 219, 479-502 (1998).
19. D. A. Jones and A. M. Stuart, Attractive invariant manifolds under approximation. Inertial manifolds, J. Differential Equations 123, 588-637 (1995).
20. U. Kirchgraber and E. Stiefel, Methoden der analytischen Störungsrechnung und ihre Anwendungen (B. G. Teubner, Stuttgart, 1978).
21. P. E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Difference Equ. Appl. 6, 32-52 (2000).
22. K. Kobayasi, Inertial manifolds for discrete approximations of evolution equations: Convergence and approximations, Adv. Math. Sci. Appl. 3, 161-189 (1993/94).
23. K. Kobayasi, Convergence and approximation of inertial manifolds for evolution equations, Differential Integral Equations 8, 1117-1134 (1995).
24. 1-approximations of inertial manifolds via finite differences, Proc. Amer. Math. Soc. 127, No. 4, 1143-1150 (1999).
25. L. Lewin, Polylogarithms and associated Functions (North-Holland, New York, 1982).
26. G. J. Lord, Attractors and inertial manifolds for finite-difference approximations of the complex Ginzburg-Landau equation, SIAM J. Numer. Anal. 34, No. 4, 1483-1512 (1997).
27. K. Nipp and D. Stoffer, Attractive invariant manifolds for maps: Existence, smoothness and continuous dependence on the map, Research Report No. 92-11, Seminar für Angewandte Mathematik, ETH Zürich (1992).
28. V. A. Pliss and G. R. Sell, Perturbations of normally hyperbolic manifolds with applications to the Navier-Stokes equations, J. Differential Equations 169, 396-492 (2001).
29. C. Pötzsche, Attractive invariant fiber bundles, Appl. Anal. 86, No. 6, 687-722 (2007).
30. C. Pötzsche, Dissipative delay endomorphisms and asymptotic equivalence, to appear in: Proceedings of the 11th International Conference on Difference Equations and Applications, Kyoto, Japan, 2006.
31. m-smoothness of invariant fiber bundles, Topol. Methods Nonlinear Anal. 24, No. 1, 107-146 (2004).
32. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences Vol. 143 (Springer, Berlin, 2002).