Advertisement

Journal of Mathematical Fluid Mechanics

, Volume 19, Issue 4, pp 623–643 | Cite as

Pullback Asymptotic Behavior of Solutions for a 2D Non-autonomous Non-Newtonian Fluid

  • Guowei LiuEmail author
Article
  • 113 Downloads

Abstract

This paper studies the pullback asymptotic behavior of solutions for the non-autonomous incompressible non-Newtonian fluid in 2D bounded domains. Firstly, with a little high regularity of the force, the semigroup method and \(\epsilon \)-regularity method are used to establish the existence of compact pullback absorbing sets. Then, with a minimal regularity of the force, by verifying the flattening property also known as the “Condition (C)”, the author proves the existence of pullback attractors for the universe of fixed bounded sets and for the another universe given by a tempered condition. Furthermore, the regularity of pullback attractors is given.

Keywords

Non-Newtonian fluid compact pullback absorbing set semigroup method \(\epsilon \)-regularity method pullback attractor flattening property 

Mathematics Subject Classification

Primary 35B41 Secondary 35Q35 76D03 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Arrieta, J.M., Carvalho, A.N.: Abstract parabolic problems with critial nonlinearities and applicaions to Navier–Stokes and heat equations. Trans. Am. Math. Soc. 352, 285–310 (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bellout, H., Bloom, F.: Incompressible bipolar and non-Newtonian viscous fluid flow. Springer, Heidelberg (2014)Google Scholar
  4. 4.
    Bellout, H., Bloom, F., Nečas, J.: Phenomenological behavior of multipolar viscous fluids. Quart. Appl. Math. 50, 559–583 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bellout, H., Bloom, F., Nečas, J.: Young measure-valued solutions for non-Newtonian incompressible viscous fluids. Commun. PDE. 19, 1763–1803 (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier–Stokes equations and Related Models. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bloom, F., Hao, W.: Regularization of a non-Newtonian system in unbounded channel: existence and uniqueness of solutions. Nonlinear Anal. 44, 281–309 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bloom, F., Hao, W.: Regularization of a non-Newtonian system in an unbounded channel: existence of a maximal compact attractor. Nonlinear Anal. 43, 743–766 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)zbMATHGoogle Scholar
  10. 10.
    Cheban, D.N.: Global Attractors of Non-Autonomous Dynamical and Control Systems, 2nd edn. World Scientific Publishing Co. Pte. Ltd, Hackensack (2015)CrossRefzbMATHGoogle Scholar
  11. 11.
    Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics, vol. 49. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  12. 12.
    Fujita, H., Kato, T.: On the Navier–Stokes initial value problem. I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    García-Luengo, J., Marín-Rubio, P., Real, J.: Pullback attractors in V for non-autonomous 2D-Navier–Stokes equations and their tempered behavior. J. Differ. Equ. 252, 4333–4356 (2012)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    García-Luengo, J., Marín-Rubio, P., Real, J., Robinson, J.C.: Pullback attractors for the non-autonomous 2D Navier–Stokes equations for minimally regular forcing. Discr. Contin. Dyn. Syst. 34, 203–227 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Guo, B., Zhu, P.: Partial regularity of suitable weak solution to the system of the incompressible non-Newtonian fluids. J. Differ. Equ. 178, 281–297 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kloeden, P.E., Langa, J.A.: Flattening, squeezing and the existence of random attractors. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463, 163–181 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kloeden, P.E., Langa, J.A., Real, J.: Pullback V-attractors of the 3-dimensional globally modified Navier–Stokes equations. Commun. Pure Appl. Anal. 6, 937–955 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach, New York (1969)zbMATHGoogle Scholar
  19. 19.
    Liu, G., Zhao, C., Cao, J.: \(H^4\)-boundedness of pullback attractor for a 2D non-Newtonian fluid flow. Front. Math. Chin. 8, 1377–1390 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Efendiev, M.: Attractors for Degenerate Parabolic Type Equations. American Mathematical Society, Providence (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Málek, J., Nečas, J., Rokyta, M., Ružička, M.: Weak and Measure-Valued Solutions to Evolutionary PDE. Champman-Hall, London (1996)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ma, Q., Wang, S., Zhong, C.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Univ. Math. J. 51, 1541–1559 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Pokorný, M.: Cauchy problem for the non-Newtonian viscous incompressible fluids. Appl. Math. 41, 169–201 (1996)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Robinson, J.C.: Infinite-Dimensional Dynamical System. Cambridge University Press, Cambridge (2001)Google Scholar
  25. 25.
    Sell, G., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  26. 26.
    Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  27. 27.
    Zhao, C., Li, Y.: \(H^2\)-compact attractor for a non-Newtonian system in two-dimensional unbound domains. Nonlinear Anal. 56, 1091–1103 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Zhao, C., Zhou, S.: Pullback attractors for a non-autonomous incompressible non-Newtonian fluid. J. Differ. Equ. 238, 394–425 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Zhao, C., Li, Y., Zhou, S.: Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid. J. Differ. Equ. 247, 2331–2363 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Zhao, C., Zhou, S., Li, Y.: Existence and regularity of pullback attractors for an incompressible non-Newtonian fluid with delays. Quart. Appl. Math. 67, 503–540 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Zhao, C., Liu, G., Wang, W.: Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors. J. Math. Fluid Mech. 16, 243–262 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Zhao, C., Liu, G., An, R.: Global well-posedness and pullback attractors for an incompressible non-Newtonian fluid with infinite delays. Differ. Equ. Dyn. Syst. doi: 10.1007/s12591-014-0231-9

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

Personalised recommendations