Arbitrarily accurate approximate inertial manifolds of fixed dimension. (English) Zbl 1052.34509

Summary: By employing an embedding result due to Ma\=né, and its recent strengthening due to Foias and Olson it is shown that a global attractor with finite fractal (box counting) dimension \(d\) lies within an arbitrarily small neighbourhood of a smooth graph over the space spanned by the first \(2d+1\) Fourier-Galerkin modes. The proof is, however, nonconstructive.


34G20 Nonlinear differential equations in abstract spaces
34D45 Attractors of solutions to ordinary differential equations
47H20 Semigroups of nonlinear operators
Full Text: DOI


[1] Hale, J.K., Asymptotic behaviour of dissipative systems, () · Zbl 0139.03501
[2] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, () · Zbl 0871.35001
[3] Mañé, R., (), 230
[4] Henry, D., Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001
[5] Foias, C.; Sell, G.R.; Temam, R., C. R. acad. sci. I, 301, 139, (1985)
[6] Chow, S.-N.; Lu, K.; Sell, G.R., J. math. anal. appl., 169, 283, (1992)
[7] Foias, C.; Manley, O.; Temam, R., Math. mod. num. anal., 22, 93, (1988)
[8] Foias, C.; Sell, G.R.; Titi, E.S., J. dyn. diff. eq., 1, 199, (1989)
[9] Debussche, A.; Temam, R., J. math. pures appl., 73, 489, (1994)
[10] Marion, M., J. dyn. diff. eq., 1, 245, (1989)
[11] Foias, C.; Olson, E., Indiana univ. math. J., 45, 603, (1996)
[12] Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501
[13] M.S. Jolly, J. Diff. Eq. 78, 220.
[14] Brunovský, P., J. dyn. diff. eq., 2, 293, (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.