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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.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34D45 Attractors of solutions to ordinary differential equations
47H20 Semigroups of nonlinear operators
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[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)
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