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September  2017, 6(3): 345-356. doi: 10.3934/eect.2017018

The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors

1. 

Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, P. O. Box 1160,41080-Sevilla, Spain

2. 

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

* Corresponding author: Alain Haraux

Received  April 2017 Revised  May 2017 Published  July 2017

We prove an estimation of the Kolmogorov $\varepsilon$-entropy in $H$ of the unitary ball in the space $V$, where $H$ is a Hilbert space and $V$ is a Sobolev-like subspace of $H$. Then, by means of Zelik's result [7], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.

Citation: María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations & Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018
References:
[1]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl, 62 (1983), 441-491. Google Scholar

[2]

Z. Chen, A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors, Chaos Solitons Fractals, 3 (1993), 575-582. doi: 10.1016/0960-0779(93)90007-N. Google Scholar

[3]

V. V. Chepyzhov and A. A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819. doi: 10.1016/S0362-546X(99)00309-0. Google Scholar

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I. DumerM. S. Pinsker and V. V. Prelov, On coverings of ellipsoids in Euclidean spaces, Transactions on Information Theory, 50 (2004), 2348-2356. doi: 10.1109/TIT.2004.834759. Google Scholar

[5]

P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318. Google Scholar

[6]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[7]

S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mem. Math. Appl., 24 (2000), 1-25. Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl, 62 (1983), 441-491. Google Scholar

[2]

Z. Chen, A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors, Chaos Solitons Fractals, 3 (1993), 575-582. doi: 10.1016/0960-0779(93)90007-N. Google Scholar

[3]

V. V. Chepyzhov and A. A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819. doi: 10.1016/S0362-546X(99)00309-0. Google Scholar

[4]

I. DumerM. S. Pinsker and V. V. Prelov, On coverings of ellipsoids in Euclidean spaces, Transactions on Information Theory, 50 (2004), 2348-2356. doi: 10.1109/TIT.2004.834759. Google Scholar

[5]

P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318. Google Scholar

[6]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[7]

S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mem. Math. Appl., 24 (2000), 1-25. Google Scholar

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