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Applied Mathematics and Mechanics

, Volume 34, Issue 2, pp 189–208 | Cite as

Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter \(H \in \left( {\tfrac{1} {4},\tfrac{1} {2}} \right)\)

  • Jin Li (李 劲)Email author
  • Jian-hua Huang (黄建华)
Article

Abstract

A two-dimensional (2D) stochastic incompressible non-Newtonian fluid driven by the genuine cylindrical fractional Brownian motion (FBM) is studied with the Hurst parameter \(H \in \left( {\tfrac{1} {4},\tfrac{1} {2}} \right)\) under the Dirichlet boundary condition. The existence and regularity of the stochastic convolution corresponding to the stochastic non-Newtonian fluids are obtained by the estimate on the spectrum of the spatial differential operator and the identity of the infinite double series in the analytic number theory. The existence of the mild solution and the random attractor of a random dynamical system are then obtained for the stochastic non-Newtonian systems with \(H \in \left( {\tfrac{1} {2},1} \right)\) without any additional restriction on the parameter H.

Key words

infinite-dimensional fractional Brownian motion (FBM) stochastic convolution stochastic non-Newtonian fluid random attractor 

Chinese Library Classification

O175.2 

2010 Mathematics Subject Classification

35Q35 35R60 60G22 37L55 

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References

  1. [1]
    Bellout, H., Bloom, F., and Nečas, J. Phenomenological behavior of multipolar viscous fluids. Quarterly of Applied Mathematics, 50(3), 559–583 (1992)MathSciNetzbMATHGoogle Scholar
  2. [2]
    Bloom, F. and Hao, W. Regularization of a non-Newtonian system in an unbounded channel: existence and uniqueness of solutions. Nonlinear Analysis: Theory, Methods & Applications, 44(3), 281–309 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Guo, B. L. and Guo, C. X. The convergence of non-Newtonian fluids to Navier-Stokes equations. Journal of Mathematical Analysis and Applications, 357(2), 468–478 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Ladyzhenskaya, O. A. The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York (1963)zbMATHGoogle Scholar
  5. [5]
    Zhao, C. D. and Duan, J. Q. Random attractor for the Ladyzhenskaya model with additive noise. Journal of Mathematical Analysis and Applications, 362(1), 241–251 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Zhao, C. D. and Zhou, S. F. Pullback attractors for a non-autonomous incompressible non-Newtonian fluid. Journal of Differential Equations, 238(2), 394–425 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Li, J. and Huang, J. H. Dynamics of 2D stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, Series B, 17(7), 2483–2508 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Kolmogorov, A. N. Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. Doklady, 26, 115–118 (1940)Google Scholar
  9. [9]
    Mandelbrot, B. B. and van Ness, J. W. Fractional Brownian motions, fractional noises and applications. SIAM Review, 10, 422–437 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Shiryaev, A. N. Essentials of stochastic finance. Advanced Series on Statistical Science & Applied Probability, Vol. 3, World Scientific Publishing Co, Inc., New Jersey (1999)Google Scholar
  11. [11]
    Daniel, H. Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin (1981)Google Scholar
  12. [12]
    Samko, S. G., Kilbas, A. A., and Marichev, O. I. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York (1993)zbMATHGoogle Scholar
  13. [13]
    Alòs, E., Mazet, O., and Nualart, D. Stochastic calculus with respect to Gaussian processes. Annals of Probability, 29(2), 766–801 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Duncan, T. E., Maslowski, B., and Pasik-Duncan, B. Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion. SIAM Journal on Applied Mathematics, 40(6), 2286–2315 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Tindel, S., Tudor, C. A., and Viens, F. Stochastic evolution equations with fractional Brownian motion. Probability Theory and Related Fields, 127(2), 186–204 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Courant, R. and Hilbert, D. Methods of Mathematical Physics, Wiley-InterScience, New York (1966)Google Scholar
  17. [17]
    Kelliher, J. P. Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane. Pacific Journal of Mathematics, 244(1), 99–132 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Borwein, J. M. and Borwein, P. B. PI and the AGM, Wiley-InterScience, New York (1987)zbMATHGoogle Scholar
  19. [19]
    Wang, G. L., Zeng, M., and Guo, B. L. Stochastic Burgers’ equation driven by fractional Brownian motion. Journal of Mathematical Analysis and Applications, 371(1), 210–222 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Decreusefond, L. and Üstünel, A. S. Stochastic analysis of the fractional Brownian motion. Potential Analysis, 10, 177–214 (1996)CrossRefGoogle Scholar
  21. [21]
    Da Prato, G. and Zabczyk, J. Ergodicity for Infinite-Dimensional Systems, Cambridge University Press, Cambridge (1996)zbMATHCrossRefGoogle Scholar
  22. [22]
    Maslowski, B. and Schmalfuss, B. Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Stochastic Analysis and Applications, 22(6), 1577–1607 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Crauel, H. and Flandoli, F. Attractors for random dynamical systems. Probability Theory and Related Fields, 100(3), 365–393 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Arnold, L. Random Dynamical Systems, Springer-Verlag, Berlin (1998)zbMATHCrossRefGoogle Scholar
  25. [25]
    Temam, R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, New York (1997)zbMATHGoogle Scholar
  26. [26]
    Temam, R. Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam (1977)zbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematics and System ScienceNational University of Defense TechnologyChangshaP. R. China

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