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September  2020, 28(3): 1343-1356. doi: 10.3934/era.2020071

The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay

School of information and Mathematics, Yangtze University, Jingzhou 434023, China

Received  February 2020 Revised  May 2020 Published  July 2020

Fund Project: The author is supported by NSFC grant No. 61673006

In this paper, two properties of the pullback attractor for a 2D non-autonomous micropolar fluid flows with delay on unbounded domains are investigated. First, we establish the $ H^1 $-boundedness of the pullback attractor. Further, with an additional regularity limit on the force and moment with respect to time t, we remark the $ H^2 $-boundedness of the pullback attractor. Then, we verify the upper semicontinuity of the pullback attractor with respect to the domains.

Citation: Wenlong Sun. The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay. Electronic Research Archive, 2020, 28 (3) : 1343-1356. doi: 10.3934/era.2020071
References:
[1]

J. ChenZ.-M. Chen and B.-Q. Dong, Existence of $H^2$-global attractors of two-dimensional micropolar fluid flows, J. Math. Anal. Appl., 322 (2006), 512-522.  doi: 10.1016/j.jmaa.2005.09.011.  Google Scholar

[2]

J. ChenZ.-M. Chen and B.-Q. Dong, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity, 20 (2007), 1619-1635.  doi: 10.1088/0951-7715/20/7/005.  Google Scholar

[3]

B.-Q. Dong and Z.-M. Chen, Global attractors of two-dimensional micropolar fluid flows in some unbounded domains, Appl. Math. Comput., 182 (2006), 610-620.  doi: 10.1016/j.amc.2006.04.024.  Google Scholar

[4]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[5]

J. García-LuengoP. Marín-Rubio and J. Real, $H^2$-boundedness of the pullback attractors for non-autonomous 2D Navier-Stokes equations in bounded domains, Nonlinear Anal., 74 (2011), 4882-4887.  doi: 10.1016/j.na.2011.04.063.  Google Scholar

[6]

G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[7]

G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Math. Comput. Modelling, 34 (2001), 487-509.  doi: 10.1016/S0895-7177(01)00078-4.  Google Scholar

[8]

G. Łukaszewicz and W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., 55 (2004), 247-257.  doi: 10.1007/s00033-003-1127-7.  Google Scholar

[9]

G. Łukaszewicz and A. Tarasińska, On $H^1$-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Anal., 71 (2009), 782-788.  doi: 10.1016/j.na.2008.10.124.  Google Scholar

[10]

A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation, IEEE Trans. Automat. Control, 29 (1984), 1058-1068.  doi: 10.1109/TAC.1984.1103436.  Google Scholar

[11]

P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799.  doi: 10.1016/j.na.2006.09.035.  Google Scholar

[12]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[13]

W. Sun, Micropolar fluid flows with delay on 2D unbounded domains, J. Appl. Anal. Comput., 8 (2018), 356-378.  doi: 10.11948/2018.356.  Google Scholar

[14]

W. Sun, J. Cheng and X. Han, Random attractors for 2D stochastic micropolar fluid flows on unbounded domains, Discrete Contin. Dyn. Syst. Ser. B. doi: 10.3934/dcdsb.2020189.  Google Scholar

[15]

W. Sun and Y. Li, Asymptotic behavior of pullback attractor for non-autonomous micropolar fluid flows in 2D unbounded domains, Electronic J. Differential Equations, 2018, 21pp.  Google Scholar

[16]

W. Sun and G. Liu, Pullback attractor for the 2D micropolar fluid flows with delay on unbounded domains, Bull. Malays. Math. Sci. Soc., 42 (2019), 2807-2833.  doi: 10.1007/s40840-018-0634-9.  Google Scholar

[17]

C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 53 (2012), 22pp. doi: 10.1063/1.4769302.  Google Scholar

[18]

C. ZhaoW. Sun and C. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dyn. Partial Differ. Equ., 12 (2015), 265-288.  doi: 10.4310/DPDE.2015.v12.n3.a4.  Google Scholar

[19]

C. ZhaoS. Zhou and X. Lian, $H^1$-uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains, Nonlinear Anal. Real World Appl., 9 (2008), 608-627.  doi: 10.1016/j.nonrwa.2006.12.005.  Google Scholar

show all references

References:
[1]

J. ChenZ.-M. Chen and B.-Q. Dong, Existence of $H^2$-global attractors of two-dimensional micropolar fluid flows, J. Math. Anal. Appl., 322 (2006), 512-522.  doi: 10.1016/j.jmaa.2005.09.011.  Google Scholar

[2]

J. ChenZ.-M. Chen and B.-Q. Dong, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity, 20 (2007), 1619-1635.  doi: 10.1088/0951-7715/20/7/005.  Google Scholar

[3]

B.-Q. Dong and Z.-M. Chen, Global attractors of two-dimensional micropolar fluid flows in some unbounded domains, Appl. Math. Comput., 182 (2006), 610-620.  doi: 10.1016/j.amc.2006.04.024.  Google Scholar

[4]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[5]

J. García-LuengoP. Marín-Rubio and J. Real, $H^2$-boundedness of the pullback attractors for non-autonomous 2D Navier-Stokes equations in bounded domains, Nonlinear Anal., 74 (2011), 4882-4887.  doi: 10.1016/j.na.2011.04.063.  Google Scholar

[6]

G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[7]

G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Math. Comput. Modelling, 34 (2001), 487-509.  doi: 10.1016/S0895-7177(01)00078-4.  Google Scholar

[8]

G. Łukaszewicz and W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., 55 (2004), 247-257.  doi: 10.1007/s00033-003-1127-7.  Google Scholar

[9]

G. Łukaszewicz and A. Tarasińska, On $H^1$-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Anal., 71 (2009), 782-788.  doi: 10.1016/j.na.2008.10.124.  Google Scholar

[10]

A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation, IEEE Trans. Automat. Control, 29 (1984), 1058-1068.  doi: 10.1109/TAC.1984.1103436.  Google Scholar

[11]

P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799.  doi: 10.1016/j.na.2006.09.035.  Google Scholar

[12]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[13]

W. Sun, Micropolar fluid flows with delay on 2D unbounded domains, J. Appl. Anal. Comput., 8 (2018), 356-378.  doi: 10.11948/2018.356.  Google Scholar

[14]

W. Sun, J. Cheng and X. Han, Random attractors for 2D stochastic micropolar fluid flows on unbounded domains, Discrete Contin. Dyn. Syst. Ser. B. doi: 10.3934/dcdsb.2020189.  Google Scholar

[15]

W. Sun and Y. Li, Asymptotic behavior of pullback attractor for non-autonomous micropolar fluid flows in 2D unbounded domains, Electronic J. Differential Equations, 2018, 21pp.  Google Scholar

[16]

W. Sun and G. Liu, Pullback attractor for the 2D micropolar fluid flows with delay on unbounded domains, Bull. Malays. Math. Sci. Soc., 42 (2019), 2807-2833.  doi: 10.1007/s40840-018-0634-9.  Google Scholar

[17]

C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 53 (2012), 22pp. doi: 10.1063/1.4769302.  Google Scholar

[18]

C. ZhaoW. Sun and C. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dyn. Partial Differ. Equ., 12 (2015), 265-288.  doi: 10.4310/DPDE.2015.v12.n3.a4.  Google Scholar

[19]

C. ZhaoS. Zhou and X. Lian, $H^1$-uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains, Nonlinear Anal. Real World Appl., 9 (2008), 608-627.  doi: 10.1016/j.nonrwa.2006.12.005.  Google Scholar

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