Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-02T01:53:38.623Z Has data issue: false hasContentIssue false
  • English
  • Français

An extension of the principle of spatial averaging for inertial manifolds

Part of: Parabolic equations and systems Qualitative theory Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  09 April 2009

Hyukjin Kwean
Hyukjin Kwean
Affiliation:
Department of Mathematics Education, College of Education, Korea University, Sungbuk-Ku Anam-Dong, 136-701 Seoul, Korea e-mail: kwean@kuccnx.korea.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

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 extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

Keywords

dissipative systemsinertial manifoldreaction diffusion equationprinciple of spatial averaging

MSC classification

Secondary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 34C29: Averaging method 35K57: Reaction-diffusion equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 1 , February 1999 , pp. 125 - 142
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Fabes, E., Luskin, M. and Sell, G. R., ‘Construction of inertial manifolds by elliptic regularization’, J. Differential Equations 89 (1991), 335–387.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. and Temam, R., ‘Some analytic and geometric properties of the solutions of Navier-Stokes equations’, J. Math. Pures Appl. 58 (1979), 339–368.Google Scholar
[4]Jolly, M., ‘Explicit construction of an inertial manifold for a reaction diffusion equation’, J. Differential Equations 78 (1991), 220–261.CrossRefGoogle Scholar
[5]Kwak, M., ‘Finite dimensional description of convective reaction diffusion equations’, J. Dynamics Differential Equations 4 (1992), 515–543.CrossRefGoogle Scholar
[6]Kwak, M., ‘Finite dimensional inertial forms for 2D Navier-Stokes equations’, Indiana Univ. Math. J. 41 (1992), 927–982.CrossRefGoogle Scholar
[7]Kwean, H., Inertial manifolds for reaction diffusion equations: an extension of the principle of spatial averaging (Dissertation, University of Minnesota, 1996).Google Scholar
[8]Mallet-Paret, J., ‘Negatively invariant sets of compact maps and an extension of a theorem of Cartwright’, J. Differential Equations 22 (1976), 331–348.CrossRefGoogle Scholar
[9]Mallet-Paret, J. and Sell, G. R., ‘Inertial manifolds for reaction diffusion equation in higher space dimensions’, J. Amer. Math. Soc. 1 (1988), 805–866.CrossRefGoogle Scholar
[10]Mañé, R., On the dimension of compact invariant sets of certain nonlinear maps, Lecture Notes in Math., 898 (Springer, New York, 1981) pp. 230–242.Google Scholar
[11]Pinsky, A. M., ‘The eigenvalues of an equilateral triangle’, SIAM J. Math. 11 (1980), 819–827.CrossRefGoogle Scholar
[12]Richards, I., ‘On the gaps between numbers which are the sum of two squares’, Adv. Math. 46 (1982), 1–2.CrossRefGoogle Scholar
[13]Temam, R., Infinite dimensional dynamical in mechanics and physics, Applied Math. Sci. 68 (Springer, New York, 1988).CrossRefGoogle Scholar