In this paper, we study the pullback asymptotic behavior of solutions for a nonautonomous incompressible non-Newtonian fluid in two-dimensional bounded domains. We prove that our system possesses pullback attractors in the space H and then obtain upper semicontinuity (in) of pullback attractors of the nonautonomous incompressible non-Newtonian fluid with the nonautonomous perturbation under some proper assumptions, i.e., the pullback attractors of the incompressible Non-Newtonian fluid withand the global attractor of (1) with for any .
:Incompressible non-Newtonian fluid; pullback attractors; pullback contracting; upper semicontinuity
Haiyan Li, Upper Semicontinuity of Pullback Attractors for a Nonautonomous Incompressible Non-Newtonian Fluid, Advances in Industrial Engineering and Management, Vol.3, No.4, 2014, pp.27-34, doi: 10.7508/AIEM-V3-N4-27-34
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 Arrieta, J., Carvalho, A. N., and Hale, J. K, 1992. A damped hyerbolic equation with critical exponent, Communications in partial differential equations, no.17, pp.841-866.
 Babin, A. V. and Vishik, M. I, 1992. Attractors of evolution equations. Elsevier, North-Holland, Amsterdam.
 Bates, P. W., Lisei, H., and Lu, K, 2006. Attractors for stochastic lattice dynamical systems. Stochastics Dyn., vol. 6, no. 1, pp.1-22.
 Bellout, H., Bloom, F., and Necas, J, 1993. Weak and measure-valued solutions for non-Newtonian fluids, CR Acad. Sci. Paris, no. 317, pp. 795-800.
 Bloom, F. and Hao, W., 2001. Regularization of a non-Newtonian system in an unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal., TMA, vol. 44, no. 3, pp. 281-309.
 Bloom, F. and Hao, W., 2001. Regularization of a non-Newtonian system in an unbounded channel: Existence and uniqueness of a maximal compact attractor, Nonlinear Anal., TMA, vol. 43, pp. 743-766.
 T. Caraballo and J. A. Langa, 2003.On the uppper semicontinuity of attractors for nonautonomous and random dynamical systems, Dyn. Contin. Discrete Impulsive Syst., Ser. A: Math. Anal., vol. 10, pp. 491-513.
 T. Caraballo, J. A. Langa, and J. C. Robinson, 1998.Uppper semicontinuity of attractors
for small random perturbations of dynamical systems, Comm. Partial Differential Equations, vol. 23, pp. 1557-1581.
 T. Caraballo, G. Lukaszewicz, and J., 2006. Real, Pullback attractors for asymptotically
compact nonautonomous dynamical systems, Nonlinear Anal., TMA, vol. 64, pp. 484-498.
 V. Chepyzhov and M. I. Vishik, 2002.Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, vol. 49.
 H. Crauel and F. Flandoli, 1994. Attractors for random dynamical systems, Probab. Theory Relat. Fields, Vol. 100, pp. 365-393.
 Hale, J. K. Asymptotic Behavior of Dissipative Systems, 1988. Amer. Math. Soc., Providence.
 Y. Li and C. Zhao, 2002. Global attractor for a non-Newtonian system in two-dimensional unbounded domains, Acta Anal. Funct. Appl., vol. 4, pp. 343-349.
 Q. F. Ma, S. H. Wang, and C. K. Zhong, 2002.Necessary and su_cient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., vol. 51, pp. 1541-1559.
 Necas, J., Málek, J., Rokyta, M., and Ruzicka, M. 1996. Weak and measure-valued solutions to evolutionary PDEs, vol. 13. CRC Press.
 Matu s̆c̆, S. and Medvid'ová-Luká c̆, M., 1998. Bipolar isothermal non-Newtonian compressible fluids., J. Math. Anal. Appl., vol. 225, 168-192.
 Nečasová, Š. and Penel,P.,2006. Incompressible non-Newtonian fluids: Time asymptotic behaviour of weak solutions, Math. Method. Appl. Sci, vol. 14, no.29, pp. 1615-1630.
 G. Sell and Y. You, 2002.Dynamics of Evolutionary Equations, Springer-Verlag, New York, vol.143.
 C. Y. Sun and M. H. Yang, 2008. Dynamics of the nonclassical diffusion equations, Asymptotic Anal., vol. 59, pp. 51-81.
 R. Temam, 1997. Infnite-Dimensional dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, New York.
 Y. H. Wang, 2008. Pullback attractors for nonautonomous wave equations with ritical exponent, Nonlinear Anal., TMA, vol. 68, pp. 365-376.
 Y. H. Wang and Y. Qin, 2010.Upper semicontinuity of pullback attractors for nonclassical diffusion equations, J. Math. Phys., vol. 51, pp, 1-10.
 Y. L. Xiao, 2002. Attractors for a noncalssical diffusion equation, Acta Math. Sinica. English Ser., vol. 18, pp. 273-276.
 S. Zelik, 2004.Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Comm. Partial Differential Equations, vol.3, pp. 921-934.
 C. Zhao and S. Li, 2004.H2-compact attractor for a non-Newtonian system in two dimensional unbounded domains, Nonlinear Anal., TMA, vol. 7, pp. 1093-1103.
 C. Zhao and S. Zhou, 2007.Pullback attractors for a non-autonomous incompressible non-
Newtonian fluid, J. Differential Equation, vol. 283, pp. 394-425.
 C. Zhao, Y. Li and S. Zhou, 2009.Regularity of trajetory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, J. Differential Equations, vol. 247, pp. 2331-2363.