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Mat. Sb., 1998, Volume 189, Number 2, Pages 81–110 (Mi msb301)  

This article is cited in 25 scientific papers (total in 25 papers)

Kolmogorov $\varepsilon$-entropy estimates for the uniform attractors of non-autonomous reaction-diffusion systems

M. I. Vishik, V. V. Chepyzhov

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: The Kolmogorov $\varepsilon$-entropy of the uniform attractor $\mathscr A$ of a family of non-autonomous reaction-diffusion systems with external forces $g(x,t)$ is studied. The external forces $g(x,t)$ are assumed to belong to some subset $\sigma$ of $C({\mathbb R};H)$, where $H=(L_2(\Omega ))^N$, that is invariant under the group of $t$-translations. Furthermore, $\sigma$ is compact in $C({\mathbb R};H)$.
An estimate for the $\varepsilon$-entropy of the uniform attractor $\mathscr A$ is given in terms of the $\varepsilon _1=\varepsilon _1(\varepsilon )$-entropy of the compact subset $\sigma_l$ of $C([0,l];H)$ consisting of the restrictions of the external forces $g(x,t)\in \sigma$ to the interval $[0,l]$, $l=l(\varepsilon )$ ($\varepsilon _1(\varepsilon )\sim \mu \varepsilon $, $l(\varepsilon )\sim \tau \log _2(1/\varepsilon )$). This general estimate is illustrated by several examples from different fields of mathematical physics and information theory.

DOI: https://doi.org/10.4213/sm301

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English version:
Sbornik: Mathematics, 1998, 189:2, 235–263

Bibliographic databases:

UDC: 517.95
MSC: 35K57, 35B40
Received: 18.09.1997

Citation: M. I. Vishik, V. V. Chepyzhov, “Kolmogorov $\varepsilon$-entropy estimates for the uniform attractors of non-autonomous reaction-diffusion systems”, Mat. Sb., 189:2 (1998), 81–110; Sb. Math., 189:2 (1998), 235–263

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. V. Zelik, “An attractor of a nonlinear system of reaction-diffusion equations in $\mathbb R^n$ and estimates for its $\epsilon$-entropy”, Math. Notes, 65:6 (1999), 790–792  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. S. V. Zelik, “The attractor of a quasilinear hyperbolic equation with dissipation in $\mathbb R^n$: Dimension and $\varepsilon$-entropy”, Math. Notes, 67:2 (2000), 248–251  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Zelik, SV, “The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's epsilon-entropy”, Mathematische Nachrichten, 232 (2001), 129  crossref  mathscinet  zmath  isi
    4. Zelik, SV, “The attractor for a nonlinear hyperbolic equation in the unbounded domain”, Discrete and Continuous Dynamical Systems, 7:3 (2001), 593  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    5. A. Mielke, S. V. Zelik, “Infinite-dimensional trajectory attractors of elliptic boundary-value problems in cylindrical domains”, Russian Math. Surveys, 57:4 (2002), 753–784  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Zelik S.V., “Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity”, Comm. Pure Appl. Math., 56:5 (2003), 584–637  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. M. I. Vishik, V. V. Chepyzhov, “Kolmogorov $\varepsilon$-Entropy in Problems on Global Attractors of Evolution Equations of Mathematical Physics”, Problems Inform. Transmission, 39:1 (2003), 2–20  mathnet  crossref  mathscinet  zmath
    8. Jia Qiuli, Zhou Shengfan, Yin Fuqi, “Kolmogorov entropy of global attractor for dissipative lattice dynamical systems”, J. Math. Phys., 44:12 (2003), 5805–5810  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    9. Efendiev M., Miranville A., Zelik S., “Infinite dimensional exponential attractors for a non-autonomous reaction-diffusion system”, Math. Nachr., 248 (2003), 72–96  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    10. Lord, GJ, “Numerical computation of epsilon-entropy for parabolic equations with analytic solutions”, Physica D-Nonlinear Phenomena, 194:1–2 (2004), 65  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    11. Efendiev, M, “Global and exponential attractors for nonlinear react ion-diffusion systems in unbounded domains”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 134 (2004), 271  crossref  mathscinet  zmath  isi
    12. Efendiev, M, “Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation”, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences, 460:2044 (2004), 1107  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    13. Efendiev, M, “Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 135 (2005), 703  crossref  mathscinet  zmath  isi  elib
    14. Zhou, SF, “Kolmogorov's epsilon-entropy of attractors for lattice systems”, International Journal of Bifurcation and Chaos, 15:7 (2005), 2295  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    15. Chueshov I., Lasiecka I., “Kolmogorov's epsilon-entropy for a class of invariant sets and dimension of global attractors for second-order evolution equations with nonlinear damping”, Control Theory of Partial Differential Equations, Pure and Applied Mathematics : A Program of Monographs, Textbooks, and Lecture Notes, 242, 2005, 51–69  crossref  mathscinet  zmath  isi
    16. Mielke A., Zelik S.V., “Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction-diffusion systems in $\mathbb R^n$”, J. Dynam. Differential Equations, 19:2 (2007), 333–389  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    17. Yin, FQ, “Global attractor for Klein-Gordon-Schrodinger lattice system”, Applied Mathematics and Mechanics-English Edition, 28:5 (2007), 695  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    18. Zelik, SV, “Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains”, Journal of Dynamics and Differential Equations, 19:1 (2007), 1  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    19. Shirikyan, A, “Euler equations are not exactly controllable by a finite-dimensional external force”, Physica D-Nonlinear Phenomena, 237:10–12 (2008), 1317  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
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    23. N. Maaroufi, “Topological entropy by unit length for the Ginzburg-Landau equation on the line”, DCDS-A, 34:2 (2013), 647  crossref  mathscinet  isi  scopus  scopus  scopus
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  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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