General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Mat. Sb.:

Personal entry:
Save password
Forgotten password?

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.


Full text: PDF file (428 kB)
References: PDF file   HTML file

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
\by M.~I.~Vishik, V.~V.~Chepyzhov
\paper Kolmogorov $\varepsilon$-entropy estimates for the~uniform attractors of non-autonomous reaction-diffusion systems
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 2
\pages 81--110
\jour Sb. Math.
\yr 1998
\vol 189
\issue 2
\pages 235--263

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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
    20. Yin, FQ, “Attractor for lattice system of dissipative Zakharov equation”, Acta Mathematica Sinica-English Series, 25:2 (2009), 321  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    21. Guo, BL, “Attractor and spatial chaos for the Brusselator in R-N”, Nonlinear Analysis-Theory Methods & Applications, 70:11 (2009), 3917  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    22. O. Goubet, N. Maaroufi, “Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework”, CPAA, 11:3 (2011), 1253  crossref  mathscinet  isi  scopus  scopus  scopus
    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
    24. Yue G.Ch., Zhong Ch.K., “Long-Term Analysis of Degenerate Parabolic Equations in R-N”, Acta. Math. Sin.-English Ser., 31:3 (2015), 383–410  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    25. Maaroufi N., “Invariance and Computation of the Extended Fractal Dimension For the Attractor of Cgl on R”, Chaos Solitons Fractals, 82 (2016), 87–96  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:605
    Full text:139
    First page:7

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020