Some closure results for inertial manifolds

Journal of Dynamics and Differential Equations

Volumn 9, Issue 3, 1997, Pages 373-400

  Robinson, J C ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Approximate inertial manifolds; Bubnov galerkin approximations; Exponential attractors; Inertial manifolds

Indexed keywords

EID: 0010534874     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1007/BF02227487     Document Type: Article

References (33)

1. Brunovský, P. (1990). The attractor of the scalar reaction diffusion equation is a smooth graph. J. Dyn. Diff. Eq. 2, 293-323.
2. Chow, S.-N., Lu, K., and Sell, G. R. (1992). Smoothness of inertial manifolds, J. Math. Anal. Appl. 169, 283-312.
3. Constantin, P., Foias, C., Nicolaenko, B., and Temam, R. (1988). Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer AMS 70, New York.
4. Devulder, C., Marion, M., and Titi, E. S. (1993). On the rate of convergence of the nonlinear Galerkin methods. Math. Comp. 60, 495-514.
5. Doering, C. R., and Gibbon, J. D. (1995). Applied Analysis of the Navier Stokes Equations, Cambridge University Press, Cambridge.
6. Eden, A., Foias, C., Nicolaenko, B., and Temam, R. (1994). Exponential Attractors for Dissipative Evolution Equations, Wiley, Chichester.
7. Falconer, K. (1985). The Geometry of Fractal Sets, Cambridge University Press, Cambridge.
8. Foias, C., Sell, G. R., and Temam, R. (1985). Variétés inertielles des équations différentielles dissipatives. C. R. Acad. Sci. Paris 1301, 139-141.
9. Foias, C., Manley, O., and Temam, R. (1988a). Modelling of the interaction of small and large eddies in two dimensional turbulent flows. Math. Mod. Numer. Anal. 22, 93-118.
10. Foias, C., Sell, G. R., and Temam, R. (1988b). Inertial manifolds for nonlinear evolution equations. J. Diff. Eq. 73, 309-353.
11. Foias, C., Nicolaenko, B., Sell, G. R., and Temam, R. (1988c). Inertial manifolds for the Kuramoto Sivashinsky equation and an estimate of their lowest dimension. J. Math. Pures Appl. 67, 197-226.
12. Foias C., Sell, G. R., and Titi, E. S. (1989). Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J. Dyn. Diff. Eq. 1, 199-244.
13. Haken, H. (1978). Synergetics, Springer, New York.
14. Hale, J. K. (1969). Ordinary Differential Equations, Wiley-Interscience, New York.
15. Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., Vol. 840, Springer-Verlag, New York.
16. Heywood, J. G., and Rannacher, R. (1993). On the question of turbulence modeling by approximate inertial manifolds and the nonlinear Galerkin method. SIAM J. Num. Anal. 30, 1603-1621.
17. Jolly, M. S. (1989). Explicit construction of an inertial manifold for a reaction diffusion equation. J. Diff. Eq. 78, 220-261.
18. Jones, D. A., Margolin, L. G., and Titi, E. S. (1995). On the effectiveness of the approximate inertial manifold - a computational study. Theoret. Comp. Fluid Dyn. 7, 243-260.
19. 1 approximations of inertial manifolds for dissipative nonlinear equations. J. Diff. Eq. 127, 54-86.
20. Marion, M., and Temam, R. (1989). Nonlinear Galerkin methods. SIAM J. Num. Anal. 26, 1139-1157.
21. Mallet-Paret, J. (1976). Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Diff. Eq. 22, 331-348.
22. Mañe, R. (1981). On the dimension of the compact invariant sets of certain nonlinear maps. Springer Lecture Notes in Math. 898, 230-242.
23. Miletta, P. D. (1994). Approximation to solutions of evolution equations. Math. Meth. Appl. Sci. 17, 753-763.
24. Naylor, W., and Sell, G. R. (1982). Linear Operator Theory in Engineering and Science, Appl. Math. Sci., Vol. 40, Springer-Verlag, New York.
25. Ninomiya, H. (1992). Some remarks on inertial manifolds. J. Math. Kyoto Univ. 32, 667-688.
26. Renardy, M., and Rogers, R. C. (1992). An Introduction to Partial Differential Equations, Springer-Verlag Texts Appl. Math., Vol. 13, New York.
27. Robinson, J. C. (1993). Inertial manifolds and the cone condition. Dyn. Syst. Appl. 2, 311-330, 451.
28. Robinson, J. C. (1995). A concise proof of the "geometric" construction of inertial manifolds. Phys. Lett. A 200, 415-417.
29. Robinson, J. C. (1996). The asymptotic completeness of inertial manifolds. Nonlinearity 9, 1325-1340.
30. Robinson, J. C. (1997). Convergent families of inertial manifolds for convergent approximations. Num. Algor. 14.
31. Sell, G. R. (1993). An optimally condition for approximate inertial manifolds. In Turbulence in Fluid Flows: A Dynamical Systems Approach, Sell, G. R., Foias, C., and Temam, R. (eds.), IMA Volumes in Mathematics and Its Applications 55, Springer, New York.
32. Temam, R. (1988). Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer AMS 68.
33. Young, W. H., and Young, G. C. (1911). On the existence of a differential coefficient. Proc. Lond. Math. Soc. 2(9), 325-335.