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Kolmogorov ε-Entropy in the Problems on Global Attractors for Evolution Equations of Mathematical Physics

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Abstract

We study the Kolmogorov ε-entropy and the fractal dimension of global attractors for autonomous and nonautonomous equations of mathematical physics. We prove upper estimates for the ε-entropy and fractal dimension of the global attractors of nonlinear dissipative wave equations.

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REFERENCES

  1. Kolmogorov, A.N. and Tikhomirov, V.M., ε-Entropy and ε-Capacity of Sets in Functional Spaces, Uspekhi Mat. Nauk., 1959, vol. 14, no. 2, pp. 3–86 [AMS Transl., Ser. 2 (Engl. Transl.), 1961, vol. 17, pp. 277–364].

    Google Scholar 

  2. Tikhomirov, V.M., On ε-Entropy of Some Classes of Analytic Functions, Dokl. Akad. Nauk SSSR, 1957, vol. 117, no. 2, pp. 191–194.

    Google Scholar 

  3. Lorenz, E.N., Deterministic Nonperiodic Flow, J. Atmosph. Sci., 1963, vol. 20, pp. 130–141.

    Google Scholar 

  4. Leonov, G.A., Lyapunov Dimension Formulas for Henon and Lorenz Attractors, Alg. Analiz, 2001, vol. 13, no. 3, pp. 1–12 [St. Petersburg Math. J. (Engl. Transl.), 2001, vol. 13, no. 3, pp. 453–464].

    Google Scholar 

  5. Babin, A.V. and Vishik, M.I., Attraktory evolyutsionnykh uravnenii, Moscow: Nauka, 1989. Translated under the title Attractors of Evolution Equations, Amsterdam: North-Holland, 1992.

    Google Scholar 

  6. Hale, J.K., Asymptotic Behavior of Dissipative Systems, Math. Surveys and Mon., vol. 25, Providence: AMS, 1988.

  7. Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Ser., vol. 68, New York: Springer, 1997, 2nd ed.

    Google Scholar 

  8. Lions, J.-L., Quelques méthodes de résolution des problémes aux limites non linéaires, Paris: Gauthier-Villars, 1969.

    Google Scholar 

  9. Babin, A.V. and Vishik, M.I., Regular Attractors of Semigroups and Evolution Equations, J. Math. Pures Appl., 1983, vol. 62, no. 4, pp. 441–491.

    Google Scholar 

  10. Ghidaglia, J.M. and Temam, R., Attractors for Damped Nonlinear Hyperbolic Equations, J. Math. Pures Appl., 1987, vol. 66, pp. 273–319.

    Google Scholar 

  11. Ladyzhenskaya, O.A., On Finding the Minimal Global Attractors for the Navier Stokes Equations and Other PDEs, Uspekhi Mat. Nauk., 1987, vol. 42, no. 6, pp. 25–60 [Russian Math. Surveys (Engl. Transl.), 1987, vol. 42, no. 6].

    Google Scholar 

  12. Chepyzhov, V.V., Vishik, M.I., and Wendland, W.L., On Nonautonomous Sine-Gordon Type Equations with a Simple Global Attractor and Some Averaging, to appear.

  13. Chepyzhov, V.V. and Vishik, M.I., Attractors for Equations of Mathematical Physics, AMS Colloq. Publ., vol. 49, Providence: AMS, 2002.

  14. Douady, A. and Oesterlé, J., Dimension de Hausdorff des attracteurs, C. R. Acad. Sci. Paris, sér. A, 1980, vol. 290, pp. 1135–1138.

    Google Scholar 

  15. Babin, A.V. and Vishik, M.I., Attractors of Evolutionary Partial Differential Equations and Estimates of Their Dimensions, Uspekhi Mat. Nauk, 1983, vol. 38, no. 4, pp. 133–187 [Russian Math. Surveys (Engl. Transl.), 1983, vol. 38, no. 4, pp. 151–213].

    Google Scholar 

  16. Constantin, P., Foias, C., and Temam, R., Attractors Representing Turbulent Flows, Mem. AMS, vol. 53, 1985.

  17. Chepyzhov, V.V. and Ilyin, A.A., On the Fractal Dimension of Invariant Sets; Applications to Navier Stokes Equations, Discr. Continuous Dynam. Syst., to appear.

  18. Blinchevskaya, M.A. and Ilyashenko, Yu.S., Estimate for the Entropy Dimension of the Maximal Attractor for k-Contracting Systems in an Infinite-Dimensional Space, Russian J. Math. Phys., 1999, vol. 6, no. 1, pp. 20–26.

    Google Scholar 

  19. Haraux, A., Systèmes dynamiques dissipatifs et applications, Paris: Masson, 1991.

    Google Scholar 

  20. Chepyzhov, V.V. and Vishik, M.I., Nonautonomous Dynamical Systems and Their Attractors, Appendix to the book: Vishik, M.I., Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge: Cambridge Univ. Press, 1992.

    Google Scholar 

  21. Chepyzhov, V.V. and Vishik, M.I., Nonautonomous Evolution Equations and Their Attractors, Russian J. Math. Phys., 1993, vol. 1, no. 2, pp. 165–190.

    Google Scholar 

  22. Chepyzhov, V.V. and Vishik, M.I., Attractors of Nonautonomous Dynamical Systems and Their Dimension, J. Math. Pures Appl., 1994, vol. 73, no. 3, pp. 279–333.

    Google Scholar 

  23. Levitan, B.M. and Zhikov, V.V., Pochti periodicheskie funktsii i differentsial'nye uravneniya, Moscow: Izd. Mosk. Univ., 1978. Translated under the title Almost Periodic Functions and Differential Equations, Cambridge: Cambridge Univ. Press, 1982.

    Google Scholar 

  24. Conway, J.H. and Sloan, N.J.A., Sphere Packing, Lattices and Groups, New York: Springer, 1988.

    Google Scholar 

  25. Vishik, M.I. and Chepyzhov, V.V., Kolmogorov ε-Entropy Estimates for the Uniform Attractors of Nonautonomous Reaction Diffusion Systems, Mat. Sbornik, 1998, vol. 189, no. 2, pp. 81–110 [Sbornik: Math. (Engl. Transl.), 1998, vol. 189, no. 2, pp. 235–263].

    Google Scholar 

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  1. Institute for Information Transmission Problems, RAS, Moscow

    M. I. Vishik & V. V. Chepyzhov

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  1. M. I. Vishik
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  2. V. V. Chepyzhov
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Vishik, M.I., Chepyzhov, V.V. Kolmogorov ε-Entropy in the Problems on Global Attractors for Evolution Equations of Mathematical Physics. Problems of Information Transmission 39, 2–20 (2003). https://doi.org/10.1023/A:1023622313456

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