# American Institute of Mathematical Sciences

October  2016, 21(8): 2687-2702. doi: 10.3934/dcdsb.2016068

## Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays

 1 Department of Mathematics, Busan National University, Busan 609-735, South Korea 2 Dong-Eui University, Department of Mathematics, 47340 Pusan, South Korea

Received  October 2015 Revised  April 2016 Published  September 2016

In this paper, we prove the existence of a pullback attractor in higher regularity space for the multi-valued process associated with the $2D$-non-autonomous incompressible non-Newtonian fluid with delays and without the uniqueness of solutions.
Citation: Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068
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##### References:
 [1] R. A. Adams, Sobolev Spaces, Acasemic Press, New York, 1975.  Google Scholar [2] H. Bellout, F. Bloom and J. Nečas, Weak and measure-valued solutions for non-Newtonian fluids, C. R. Acad. Sci. Paris, 317 (1993), 795-800.  Google Scholar [3] H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible viscous fluids, Comm. Partial Differential Equations, 19 (1994), 1763-1803. doi: 10.1080/03605309408821073.  Google Scholar [4] F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3.  Google Scholar [5] M. Boukrouche, G. Lukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, Internat. J. Engrg. Sci., 44 (2006), 830-844. doi: 10.1016/j.ijengsci.2006.05.012.  Google Scholar [6] B. T. Caraballo and J. A. Langa, Attractors for differential equations with variable delay, J. Math. Anal. Appl., 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464.  Google Scholar [7] B. T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.  Google Scholar [8] T. Caraballo and J. Real, Attractors for $2D$-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012.  Google Scholar [9] P. E. Kloedean and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141-152. doi: 10.1023/A:1019156812251.  Google Scholar [10] J. A. Langa, G. Lukaszewicz and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, Nonlinear Anal., 66 (2007), 735-749. doi: 10.1016/j.na.2005.12.017.  Google Scholar [11] X. Liu and Y. Wang, Pullback attractors for nonautonomous $2D$-Navier Stokes models with variable delays, Abstr. Appl. Anal., 2003 (2003), Art. ID 425031, 10 pp.  Google Scholar [12] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. Google Scholar [13] M. Pokorny, Cauchy problem for the non-Newtonian viscous incompressible fluids, Appl. Math., 41 (1996), 169-201.  Google Scholar [14] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, NY. USA, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [15] Y. J. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370. doi: 10.3934/dcds.2014.34.4343.  Google Scholar [16] Y. Wang and S. Zhou, Kernel sections on multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Appl. Math., 67 (2009), 343-378. doi: 10.1090/S0033-569X-09-01150-0.  Google Scholar [17] C. Zhao and Y. Li, $H^{2}$-compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Anal., 56 (2004), 1091-1103. doi: 10.1016/j.na.2003.11.006.  Google Scholar [18] C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001.  Google Scholar
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