The existence of global attractors for 2D Navier-Stokes equations in H k spaces

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Abstract

In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in H k (Ω, R 2) for any k ≥ 1, which attracts any bounded set of H k (Ω, R 2) in the H k -norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong’s conclusion.

Keywords

Navier-Stokes equations semigroup of operator global attractor regularity of attractor 

MR(2000) Subject Classification

35B40 35B41 
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References

  1. [1]
    Foias, C., Temam, R.: Structure of the set of stationary solutions of the Navier-Stokes equations. Comm. Pure Appl. Math., 30, 149–164 (1977)MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Lu, S., Wu, H., Zhong, C. K.: Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces. Dist. Cont. Dyna. Syst., 13(3), 701–719 (2005)MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of a class of pattern formation equations. Commun. Partial Differential Equations, 14, 245–297 (1989)MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, New York, Springer-Verlag, 1997MATHGoogle Scholar
  5. [5]
    Ma, Q. F., Wang, S. H., Zhong, C. K.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Univ. Math. J, 51(6), 1541–1559 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Zhong, C. K., Yang, M., Sun, C.: The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equation. J.D.E., 367–399 (2005)Google Scholar
  7. [7]
    Ma, T., Wang, S. H.: Bifurcation Theory and Application, Singapore, World Scientific, 2005Google Scholar
  8. [8]
    Ma, T., Wang, S. H.: Stability and Bifurcation of Nonlinear Evolution Equations, Beijing Academic Press, China (in Chiness), 2006Google Scholar
  9. [9]
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations, New York, Springer-Verlag, 1983MATHGoogle Scholar
  10. [10]
    Ball, J. M.: Strongly continuous semigroups, weak solutions and the variation of constants formula. Proc. Amer. Math. Soc., 63, 370–373 (1977)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Faculty of ScienceXi’an Jiaotong UniversityXi’anP. R. China
  2. 2.School of ScienceChang’an UniversityXi’anP. R. China
  3. 3.Mathematical CollegeSi’chuan UniversityChengduP. R. China

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