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New explicit and exact traveling wave solutions of (3+1)-dimensional KP equation
Global attractors of the 3D micropolar equations with damping term
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 |
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 $.
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. |
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