Advertisement

Dimensions of attractors for discretizations for Navier-Stokes equations

Article

Abstract

In this paper, we discretize the 2-D incompressible Navier-Stokes equations with the periodic boundary condition by the finite difference method. We prove that with a shift for discretization, the global solutions exist. After proving some discrete Sobolev inequalities in the sense of finite differences, we prove the existence of the global attractors of the discretized system, and we estimate the upper bounds for the Hausdorff and the fractal dimensions of the attractors. These bounds are indepent of the mesh sizes and are considerably close to those of the continuous version.

Key words

Navier-Stokes equation finite difference attractor Hausdorff dimension 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    R. Adams,Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
  2. [B-K]
    G. L. Browning and H.-O. Kreiss, Comparison of numerical methods for the calculation of two-dimensional turbulence.Math. Comp. 50(186), 369–388 (1989).Google Scholar
  3. [C-F1]
    P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2-D Navier-Stokes equations.Commun. Pure Appl. Math. 38, 1–27 (1985).Google Scholar
  4. [C-F2]
    P. Constantin and C. Foias,Navier-Stokes Equations, University of Chicago, 1988.Google Scholar
  5. [C-F-T]
    P. Constantin, C. Foias, and R. Temam, Attractors representing turbulent flows.Mem. Am. Math. Soc. 53, No. 314 (1983).Google Scholar
  6. [F]
    A. Friedman,Partial Differential Equations, Robert E. Krieger, New York, 1976.Google Scholar
  7. [G]
    J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations.C.R. Acad. Sci. Paris Ser. I Math. 305, 291–294 (1987).Google Scholar
  8. [H-L-R]
    J. Hale, X. Lin, and G. Raugel, Upper semicontinuity of attractors for approxima-tions of semigroups and partial differential equations.Math. Comp. 50(181) 89–123 (1988).Google Scholar
  9. [H]
    D. Henry, Goemetric theory of semilinear parabolic equations.Lect. Notes Math. (1981).Google Scholar
  10. [K]
    T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966.Google Scholar
  11. [L1]
    O. Ladyzhenskaya,Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.Google Scholar
  12. [L2]
    O. Ladyzhenskaya, On the dynamical systems generated by the Navier-Stokes equations (English translation).J. Sov. Math. 3, 458–479 (1972).Google Scholar
  13. [L-T]
    E. Lieb and W. Thirring,Inequalities for the Moments of Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities, Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann, Princeton Univer-sity, Princeton, NJ, 1976, pp. 269–303.Google Scholar
  14. [N]
    L. Nirenberg, On elliptic partial differential equations.Ann. Scoula Normala Superiore Pisa XIII (1959).Google Scholar
  15. [P]
    A. Papoulis,The Furier Integral and Its Applications, McGraw-Hill, New York, 1962, pp. 25–29.Google Scholar
  16. [S-W]
    E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton University, Princeton, NJ, 1971.Google Scholar
  17. [T1]
    R. Temam,Navier-Stokes Equations: Theory and Numerical Analysis, North Holland, Amsterdam, 1977.Google Scholar
  18. [T2]
    R. Temam,Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1988.Google Scholar
  19. [T3]
    R. Temam,Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.Google Scholar
  20. [V-B]
    R. Vichnevetsky and J. Bowles,Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM, Philadelphia, 1982.Google Scholar
  21. [Y]
    Y. Yan, Attractors and dimensions for a Schrödinger equation and the Sine-Gordon equation, AHPCRC Preprint 91-64 (1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Yin Van
    • 1
  1. 1.AHPCRCUniversity of MinnesotaMinneapolis

Personalised recommendations