American Institute of Mathematical Sciences

Advanced
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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015
[2]

Wenru Huo, Aimin Huang. The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2531-2550. doi: 10.3934/dcdsb.2016059
[3]

Chunxiang Zhao, Chunyan Zhao, Chengkui Zhong. The global attractor for a class of extensible beams with nonlocal weak damping. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 935-955. doi: 10.3934/dcdsb.2019197
[4]

Jingrui Su. Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3423-3434. doi: 10.3934/dcds.2017145
[5]

Xiangqing Zhao, Bing-Yu Zhang. Global controllability and stabilizability of Kawahara equation on a periodic domain. Mathematical Control and Related Fields, 2015, 5 (2) : 335-358. doi: 10.3934/mcrf.2015.5.335
[6]

Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure and Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
[7]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[8]

Bo-Qing Dong, Jiahong Wu, Xiaojing Xu, Zhuan Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4133-4162. doi: 10.3934/dcds.2018180
[9]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[10]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215
[11]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[12]

Yuncheng You. Global attractor of the Gray-Scott equations. Communications on Pure and Applied Analysis, 2008, 7 (4) : 947-970. doi: 10.3934/cpaa.2008.7.947
[13]

Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure and Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923
[14]

Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781
[15]

Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164
[16]

Xin Zhong. Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021296
[17]

Xin Zhong. Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum. Communications on Pure and Applied Analysis, 2022, 21 (2) : 493-515. doi: 10.3934/cpaa.2021185
[18]

Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575
[19]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343
[20]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253

 Impact Factor: 

Tools

Metrics

  • PDF downloads (139)
  • HTML views (235)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy