Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T02:59:02.381Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction of dimension of approximate intertial manifolds by symmetry

Published online by Cambridge University Press:  17 April 2009

Anibal Rodriguez-Bernal  and
Bixiang Wang
Anibal Rodriguez-Bernal
Affiliation:
Departmento de Matematica Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain
Bixiang Wang
Affiliation:
Departmento de Matematica Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain and Department of Applied Mathematics, Tsinghua University, Beijing 100084, China
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we study approximate inertial manifolds for nonlinear evolution partial differential equations which possess symmetry. The relationship between symmetry and dimensions of approximate inertial manifolds is established. We demonstrate that symmetry can reduce the dimensions of an approximate inertial manifold. Applications for concrete evolution equations are given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 2 , October 1999 , pp. 319 - 330
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Foias, C., Jolly, M.S., Kevrekidis, I.G., Sell, G.R. and Titi, E.S., ‘On the computation of inertial manifolds’, Physics Lett. A 131 (1988), 433–436.CrossRefGoogle Scholar
[2]Foias, C., Sell, G.R. and Temam, R., ‘Inertial manifolds for nonlinear evolutionary equations’, J. Differential Equations 73 (1988), 309–353.CrossRefGoogle Scholar
[3]Foias, C., Sell, G.R. and Titi, E.S., ‘Exponential tracking and approximation of inertial manifolds for dissipative equations’, J. Dynamics Differential Eqations 1 (1989), 199–244.CrossRefGoogle Scholar
[4]Hale, J.K., Asymptotic behaviour of dissipative systems, Math. Surveys and Monographs 25 (American Mathematical Society, Providence, R.I., 1988).Google Scholar
[5]Il′yashenko, Ju., ‘Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation’, J. Dynamics Differential Equations 4 (1992), 585–615.CrossRefGoogle Scholar
[6]Jolly, M.S., Kevrekidis, I.G. and Titi, E.S., ‘Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations’, Phys. D. 44 (1990), 38–60.CrossRefGoogle Scholar
[7]Nicolaenko, B., Sheurer, B. and Temam, R., ‘Some global dynamical properties of the Kuramoto-Sivashinsky equation: nonlinear stability and attractors’, Phys. D. 16 (1985). 155–183.CrossRefGoogle Scholar
[8]Nicolaenko, B., Sheurer, B. and Temam, R., ‘Dynamical properties of a class of pattern formation equations’. Comm. Partial Differential Eqations 14 (1989), 245–297.CrossRefGoogle Scholar
[9]Rodriguez-Bernal, A., ‘Inertial manifolds for dissipative semiflows in Banach spaces’, Appl. Anal. 37 (1990), 95–141.CrossRefGoogle Scholar
[10]Rodriguez-Bernal, A., ‘On the construction of inertial manifolds under symmetry constraints I: abstract results’, Nonlinear Anal. 19 (1992), 687–700.CrossRefGoogle Scholar
[11]Rodriguez-Bernal, A., ‘On the construction of inertial manifolds under symmetry constraints II: O(2) constraint and inertial manifolds on thin domains’, J. Math. Pures Appl. 72 (1993), 57–79.Google Scholar
[12]Sell, G.R., ‘An optimality condition for approximate inertial manifolds’, in Turbulence in fluid flows: A dynamical system approach (Springer-Verlag, Berlin, Heidelberg, New York, 1993), pp. 165–186.CrossRefGoogle Scholar
[13]Stuart, A.M., ‘Perturbation theory for infinite dimensional dynamical systems’, in Theory and numerics of ordinary and partial differential equations, (Ainsworth, M. et al. , Editors) (Oxford University Press, 1995).Google Scholar
[14]Temam, R., Infinite dimensional dynamical systems in mechanics and physics. (Springer-Verlag, Berlin, Heidelberg, New York, 1988).CrossRefGoogle Scholar