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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

María Anguiano 1, and  Alain Haraux 2,,

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.

Keywords: Fractal dimension, attractors, entropy.
Mathematics Subject Classification: 37L30, 35B41.
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

[4]

I. Dumer, M. 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. Dumer, M. 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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