# American Institute of Mathematical Sciences

August  2017, 22(6): 2339-2350. doi: 10.3934/dcdsb.2017101

## On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ

 1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA 2 Department of Mathematics, Texas A & M University, College Station, TX 77843, USA

Received  June 2016 Revised  December 2016 Published  March 2017

One particular metric that generates the weak topology on the weak global attractor $\mathcal{A}_w$ of three dimensional incompressible Navier-Stokes equations is introduced and used to obtain an upper bound for the Kolmogorov entropy of $\mathcal{A}_w$. This bound is expressed explicitly in terms of the physical parameters of the fluid flow.

Citation: Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101
##### References:
 [1] A. Biswas, C. Foias and A. Larios, On the attractor for the semi-dissipative Boussinesq equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Elsevier, 34 (2017), 381-405, arXiv: 1507.00080. [2] P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, Communications on Pure and Applied Mathematics, 38 (1985), 1-27. [3] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago Lectures in Mathematics, 1988. [4] R. M. Dudley, Metric entropy and the central limit theorem in C(S) Ann. Inst. Fourier (Grenoble), 24 (1974), 49-60. [5] C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2001. [6] C. Foias, C. Mondaini and B. Zhang, On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations: Ⅱ, In preparation. [7] C. Foias, C. Mondaini and B. Zhang, Remarks on the Weak Global Attractor of 3D NavierStokes Equations, In preparation. [8] C. Foias, R. Rosa and R. Temam, Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations, Discrete and Continuous Dynamical System, 27 (2010), 1611-1631. [9] C. Foias and J. C. Saut, Asymptotic behavior, as t→ ∞ of solutions of Navier-Stokes equations and nonlinear spectral manifolds, Indiana University Mathematics Journal, 33 (1984), 459-477. [10] C. Foias and J. C. Saut, Asymptotic integration of Navier-Stokes equations with potential forces. Ⅰ, Indiana Univ. Math. J, 40 (1990), 305-320. [11] C. Foias and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, in Directions in Partial Differential Equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wilsconsin, 54, Academic Press, Boston, MA, 54 (1987), 55-73. [12] C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J.Maht.Pures et Appl., 58 (1979), 339-368. [13] A. N. Kolmogorov, On certain asymptotic characteristics of completely bounded metric spaces, Doki.Akad.Naus SSSR, 108 (1956), 385-388. [14] A. N. Kolmogorov, The representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Doklady Akademii Nauk SSSR, 114 (1957), 953-956. [15] A. N. Kolmogorov and V. M. Tikhomirov, $\epsilon$-entropy and $\epsilon$-capacity of sets in functional spaces, Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 277-364. [16] S. Liu and B. Li, The functional dimension of some classes of spaces, Chin. Ann. Math., 26 (2005), 67-74. [17] R. Temam, Naviers-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematical, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. [18] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, Springer-Verlag, New York, 68,1997. [19] V. M. Tikhomirov, On $\epsilon$-entropy of classes of analytic functions, Dokl. Akad. Nauk SSSR, 117 (1957), 191-194. [20] V. M. Tikhomirov, Approximation theory in the twentieth century, In Mathematical Events of the Twentieth Century. Springer, Berlin, (2006), 409-436.

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##### References:
 [1] A. Biswas, C. Foias and A. Larios, On the attractor for the semi-dissipative Boussinesq equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Elsevier, 34 (2017), 381-405, arXiv: 1507.00080. [2] P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, Communications on Pure and Applied Mathematics, 38 (1985), 1-27. [3] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago Lectures in Mathematics, 1988. [4] R. M. Dudley, Metric entropy and the central limit theorem in C(S) Ann. Inst. Fourier (Grenoble), 24 (1974), 49-60. [5] C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2001. [6] C. Foias, C. Mondaini and B. Zhang, On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations: Ⅱ, In preparation. [7] C. Foias, C. Mondaini and B. Zhang, Remarks on the Weak Global Attractor of 3D NavierStokes Equations, In preparation. [8] C. Foias, R. Rosa and R. Temam, Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations, Discrete and Continuous Dynamical System, 27 (2010), 1611-1631. [9] C. Foias and J. C. Saut, Asymptotic behavior, as t→ ∞ of solutions of Navier-Stokes equations and nonlinear spectral manifolds, Indiana University Mathematics Journal, 33 (1984), 459-477. [10] C. Foias and J. C. Saut, Asymptotic integration of Navier-Stokes equations with potential forces. Ⅰ, Indiana Univ. Math. J, 40 (1990), 305-320. [11] C. Foias and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, in Directions in Partial Differential Equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wilsconsin, 54, Academic Press, Boston, MA, 54 (1987), 55-73. [12] C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J.Maht.Pures et Appl., 58 (1979), 339-368. [13] A. N. Kolmogorov, On certain asymptotic characteristics of completely bounded metric spaces, Doki.Akad.Naus SSSR, 108 (1956), 385-388. [14] A. N. Kolmogorov, The representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Doklady Akademii Nauk SSSR, 114 (1957), 953-956. [15] A. N. Kolmogorov and V. M. Tikhomirov, $\epsilon$-entropy and $\epsilon$-capacity of sets in functional spaces, Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 277-364. [16] S. Liu and B. Li, The functional dimension of some classes of spaces, Chin. Ann. Math., 26 (2005), 67-74. [17] R. Temam, Naviers-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematical, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. [18] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, Springer-Verlag, New York, 68,1997. [19] V. M. Tikhomirov, On $\epsilon$-entropy of classes of analytic functions, Dokl. Akad. Nauk SSSR, 117 (1957), 191-194. [20] V. M. Tikhomirov, Approximation theory in the twentieth century, In Mathematical Events of the Twentieth Century. Springer, Berlin, (2006), 409-436.
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