American Institute of Mathematical Sciences

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May  2021, 4(2): 117-130. doi: 10.3934/mfc.2021007

Global attractors of the 3D micropolar equations with damping term

Xiaojie Yang 1, , Hui Liu 1,, and  Chengfeng Sun 2,

1. 

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
2. 

School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, China

* Corresponding author: Hui Liu

Received  September 2020 Revised  January 2021 Published  May 2021 Early access  May 2021

Fund Project: The second author is supported by the National Natural Science Foundation of China (No. 11901342), Postdoctoral Innovation Project of Shandong Province (No. 202003040) and China Postdoctoral Science Foundation (No. 2019M652350) and the Natural Science Foundation of Shandong Province (No. ZR2018QA002). The third author is supported by the NSF of China (No. 11701269)

The 3D micropolar system with a damping term is considered by the uniform estimates. In this paper, global attractors of the 3D micropolar equations with damping term are proved for $ 3<\beta<5 $ with any $ \sigma>0 $.

Keywords: Micropolar equations, damping, global attractor, Laplacian operator, periodic domain.
Mathematics Subject Classification: Primary: 35A01, 35B41; Secondary: 35Q35.
Citation: Xiaojie Yang, Hui Liu, Chengfeng Sun. Global attractors of the 3D micropolar equations with damping term. Mathematical Foundations of Computing, 2021, 4 (2) : 117-130. doi: 10.3934/mfc.2021007
References:
[1]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.  Google Scholar

[2]

Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724.  doi: 10.1016/j.jde.2011.09.035.  Google Scholar

[3]

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

[4]

V. A. Galaktionov, On blow-up "twistors" for the Navier-Stokes equations in $\mathbb{R}^{3}$: A view from reaction-diffusion theory, preprint, arXiv: 0901.4286v1. Google Scholar

[5]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[6]

K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161.  doi: 10.1016/j.jde.2017.08.001.  Google Scholar

[7]

Y. Jia, X. Zhang and B.-Q. Dong, The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping, Nonlinear Anal. Real World Appl., 12 (2011), 1736-1747.  doi: 10.1016/j.nonrwa.2010.11.006.  Google Scholar

[8]

H. Liu and H. Gao, Decay of solutions for the 3D Navier-Stokes equations with damping, Appl. Math. Lett., 68 (2017), 48-54.  doi: 10.1016/j.aml.2016.11.013.  Google Scholar

[9]

H. Liu and H. Gao, Ergodicity and dynamics for the stochastic 3D Navier-Stokes equations with damping, Commun. Math. Sci., 16 (2018), 97-122.  doi: 10.4310/CMS.2018.v16.n1.a5.  Google Scholar

[10]

H. Liu, C. Sun and F. Meng, Global well-posedness of the 3D magneto-micropolar equations with damping, Appl. Math. Lett., 94 (2019), 38-43.  doi: 10.1016/j.aml.2019.02.026.  Google Scholar

[11]

H. Liu, C. Sun and J. Xin, Attractors of the 3D magnetohydrodynamics equations with damping, Bull. Malays. Math. Sci. Soc., 44 (2021), 337-351.  doi: 10.1007/s40840-020-00949-0.  Google Scholar

[12]

H. B. de Oliveira, Existence of weak solutions for the generalized Navier-Stokes equations with damping, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 797-824.  doi: 10.1007/s00030-012-0180-3.  Google Scholar

[13]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.  Google Scholar

[14]

X.-L. Song and Y.-R. Hou, Attractors for the three-diemensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252.  doi: 10.3934/dcds.2011.31.239.  Google Scholar

[15]

X.-L. Song and Y.-R. Hou, Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351.  doi: 10.1016/j.jmaa.2014.08.044.  Google Scholar

[16]

E. S. Titi and S. Trabelsi, Global well-posedness of a 3D MHD model in porous media, J. Geom. Mech., 11 (2019), 621-637.  doi: 10.3934/jgm.2019031.  Google Scholar

[17]

N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Math. Methods Appl. Sci., 28 (2005), 1507-1526.  doi: 10.1002/mma.617.  Google Scholar

[18]

Z. Ye, Global existence of solution to the 3D micropolar equations with a damping term, Appl. Math. Lett., 83 (2018), 188-193.  doi: 10.1016/j.aml.2018.04.002.  Google Scholar

[19]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029.  Google Scholar

show all references

References:
[1]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.  Google Scholar

[2]

Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724.  doi: 10.1016/j.jde.2011.09.035.  Google Scholar

[3]

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

[4]

V. A. Galaktionov, On blow-up "twistors" for the Navier-Stokes equations in $\mathbb{R}^{3}$: A view from reaction-diffusion theory, preprint, arXiv: 0901.4286v1. Google Scholar

[5]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[6]

K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161.  doi: 10.1016/j.jde.2017.08.001.  Google Scholar

[7]

Y. Jia, X. Zhang and B.-Q. Dong, The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping, Nonlinear Anal. Real World Appl., 12 (2011), 1736-1747.  doi: 10.1016/j.nonrwa.2010.11.006.  Google Scholar

[8]

H. Liu and H. Gao, Decay of solutions for the 3D Navier-Stokes equations with damping, Appl. Math. Lett., 68 (2017), 48-54.  doi: 10.1016/j.aml.2016.11.013.  Google Scholar

[9]

H. Liu and H. Gao, Ergodicity and dynamics for the stochastic 3D Navier-Stokes equations with damping, Commun. Math. Sci., 16 (2018), 97-122.  doi: 10.4310/CMS.2018.v16.n1.a5.  Google Scholar

[10]

H. Liu, C. Sun and F. Meng, Global well-posedness of the 3D magneto-micropolar equations with damping, Appl. Math. Lett., 94 (2019), 38-43.  doi: 10.1016/j.aml.2019.02.026.  Google Scholar

[11]

H. Liu, C. Sun and J. Xin, Attractors of the 3D magnetohydrodynamics equations with damping, Bull. Malays. Math. Sci. Soc., 44 (2021), 337-351.  doi: 10.1007/s40840-020-00949-0.  Google Scholar

[12]

H. B. de Oliveira, Existence of weak solutions for the generalized Navier-Stokes equations with damping, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 797-824.  doi: 10.1007/s00030-012-0180-3.  Google Scholar

[13]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.  Google Scholar

[14]

X.-L. Song and Y.-R. Hou, Attractors for the three-diemensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252.  doi: 10.3934/dcds.2011.31.239.  Google Scholar

[15]

X.-L. Song and Y.-R. Hou, Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351.  doi: 10.1016/j.jmaa.2014.08.044.  Google Scholar

[16]

E. S. Titi and S. Trabelsi, Global well-posedness of a 3D MHD model in porous media, J. Geom. Mech., 11 (2019), 621-637.  doi: 10.3934/jgm.2019031.  Google Scholar

[17]

N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Math. Methods Appl. Sci., 28 (2005), 1507-1526.  doi: 10.1002/mma.617.  Google Scholar

[18]

Z. Ye, Global existence of solution to the 3D micropolar equations with a damping term, Appl. Math. Lett., 83 (2018), 188-193.  doi: 10.1016/j.aml.2018.04.002.  Google Scholar

[19]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029.  Google Scholar

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