Communications in Mathematical Physics

, Volume 158, Issue 2, pp 327–339 | Cite as

A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations

  • Vincent Xiaosong Liu


For a special class of the Navier-Stokes equations on the two-dimensional torus, we give a lower bound in the formG2/3 (whereG is the Grashof number) for the Hausdorff dimension of its global attractor which is optimal up to a logarithmic term.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Special Class 
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  1. 1.
    Arnold, V.I., et al.: Some unsolved problems in the theory of differential equations and mathematics physics. Russ. Math. Surv.44:4, 157–171 (1989)Google Scholar
  2. 2.
    Arnold, V.I.: Kolmogorov's hydrodynamic attractors. Proc. R. Soc. Lond. A434, 19–22 (1991)Google Scholar
  3. 3.
    Babin, A.V., Vishik, M.I.: Attractors of partial differential evolution equations and estimates of their dimension. Russ. Math. Surv.38, 151–213 (1983)Google Scholar
  4. 4.
    Constantin, P., Foias, C.: Navier-Stokes equations. Chicago, IL: The University of Chicago Press, 1988Google Scholar
  5. 5.
    Constantin, P., Foias, C., Temam, R.: On the dimension of the attractors in two-dimensional turbulence. Physica D.30, 284–296 (1988)Google Scholar
  6. 6.
    Ghidaglia, J.-M., Temam, R.: Lower bound on the dimension of the attractor for the Navier-Stokes equations in space dimension 3. In: Mechanics, Analysis and Geometry: 200 Years after Lagrange, M. Francaviglia, D. Holms, eds., Amsterdam: ElsevierGoogle Scholar
  7. 7.
    Jones, W.B., Thron, W.J.: Continued fractions, analytic theory and applications. Encyclopedia of Math. and Its Appl., Vol.11, 1980Google Scholar
  8. 8.
    Liu, V.X.: An example of instability for the Navier-Stokes equations on the 2-dimensional torus. Comm. P.D.E.17, Nos. 11 & 12, 1995–2012 (1992)Google Scholar
  9. 9.
    Liu, V.X.: Instability for the Navier-Stokes equations on the 2-dimensional torus and a lower bound for the Hausdorff dimension of their global attractors. Commun. Math. Phys.147, 217–230 (1992)Google Scholar
  10. 10.
    Marchioro, C.: An example of absence of turbulence for any Reynolds number. Commun. Math. Phys.105, 99–106 (1986)Google Scholar
  11. 11.
    Meshalkin, L.D., Sinai, Ya. G.: Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid. J. Appl. Math. Mech.25 (1961)Google Scholar
  12. 12.
    Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal.7, 487–513 (1971)Google Scholar
  13. 13.
    Sattinger, D.H.: The mathematical problem of hydrodynamic stability. J. Math. and Mech.19, No. 9 (1970)Google Scholar
  14. 14.
    Smale, S.: Dynamics retrospective: great problems, attempts that failed. For “Non-linear Science: The Next Decade,” Los Alamos, May 1990Google Scholar
  15. 15.
    Temam, R.: Navier-Stokes equational dynamics and nonlinear functional analysis. Philadelphia: SIAM, 1983.Google Scholar
  16. 16.
    Temam, R.: Infinite dimensional dynamics systems in mechanics and physics. Berlin, Heidelberg, New York: Springer 1988Google Scholar
  17. 17.
    Yodovich, V.I.: Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. J. Appl. Math. Mech.29 (1965)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Vincent Xiaosong Liu
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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