Global stability solution of the 2D incompressible anisotropic magneto-micropolar fluid equations

Ru Bai; Tiantian Chen; Sen Liu; Ru Bai; Tiantian Chen; Sen Liu

doi:10.3934/math.20221131

AIMS Mathematics

2022, Volume 7, Issue 12: 20627-20644. doi: 10.3934/math.20221131

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Global stability solution of the 2D incompressible anisotropic magneto-micropolar fluid equations

1.
Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China
2.
College of Mathematics and Physics, Chengdu University of Technology, Chengdu 610059, China

Received: 06 July 2022 Revised: 15 September 2022 Accepted: 16 September 2022 Published: 23 September 2022
MSC : 35A05, 35Q35, 76D03

In this paper, we consider the two dimensional incompressible anisotropic magneto-micropolar fluid equations with partial mixed velocity dissipations, magnetic diffusion and horizontal vortex viscosity, and analyze the stability near a background magnetic field. At present, major works on the equations of magneto-micropolar fluid mainly focus on the global regularity of the solutions. While the stability of the solutions remains an open problem. This paper concentrates on establishing the stability for the linear and nonlinear system respectively. Two goals have been achieved. The first is to obtain the explicit decay rates for the solution of the linear system in $ H^s({\mathbb{R}}^2) $ Sobolev space. The second assesses the nonlinear stability by establishing the a priori estimate and employing bootstrapping arguments. Our results reveal that any perturbations near a background magnetic field is globally stable in Sobolev space $ H^2({\mathbb{R}}^2) $.
- 2D magneto-micropolar fluid,
- partial dissipation,
- stability,
- decay rates
Citation: Ru Bai, Tiantian Chen, Sen Liu. Global stability solution of the 2D incompressible anisotropic magneto-micropolar fluid equations[J]. AIMS Mathematics, 2022, 7(12): 20627-20644. doi: 10.3934/math.20221131

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Abstract

In this paper, we consider the two dimensional incompressible anisotropic magneto-micropolar fluid equations with partial mixed velocity dissipations, magnetic diffusion and horizontal vortex viscosity, and analyze the stability near a background magnetic field. At present, major works on the equations of magneto-micropolar fluid mainly focus on the global regularity of the solutions. While the stability of the solutions remains an open problem. This paper concentrates on establishing the stability for the linear and nonlinear system respectively. Two goals have been achieved. The first is to obtain the explicit decay rates for the solution of the linear system in $ H^s({\mathbb{R}}^2) $ Sobolev space. The second assesses the nonlinear stability by establishing the a priori estimate and employing bootstrapping arguments. Our results reveal that any perturbations near a background magnetic field is globally stable in Sobolev space $ H^2({\mathbb{R}}^2) $.

References

[1]	B. Yuan, Y. Qiao, Global regularity of 2D Leray-alpha regularized incompressible magneto-micropolar equations, J. Math. Anal. Appl., 474 (2019), 492–512. https://doi.org/10.1016/j.jmaa.2019.01.057 doi: 10.1016/j.jmaa.2019.01.057
[2]	J. Fan, Y. Zhou, Global solutions to the incompressible magneto-micropolar system in a bounded domain in 2D, Appl. Math. Lett., 118 (2021), 107125. https://doi.org/10.1016/j.aml.2021.107125 doi: 10.1016/j.aml.2021.107125
[3]	L. Ma, On two-dimensional incompressible magneto-micropolar system with mixed partial viscosity, Nonlinear Anal-Real, 40 (2018), 95–129. https://doi.org/10.1016/j.nonrwa.2017.08.014 doi: 10.1016/j.nonrwa.2017.08.014
[4]	Y. Guo, H. Shang, Global well-posedness of two-dimensional magneto-micropolar equations with partial dissipation, Appl. Math. Comput., 313 (2017), 392–407. https://doi.org/10.1016/j.amc.2017.06.017 doi: 10.1016/j.amc.2017.06.017
[5]	H. Lin, S. Li, Global well-posedness for the 2$ \frac 12$D incompressible magneto-micropolar fluid equations with mixed partial viscosity, Comput. Math. Appl., 72 (2016), 1066–1075. https://doi.org/10.1016/j.camwa.2016.06.028 doi: 10.1016/j.camwa.2016.06.028
[6]	Y. Lin, S. Li, Global well-posedness for magneto-micropolar system in 2$ \frac 12$ dimensions, Appl. Math. Comput., 280 (2016), 72–85. https://doi.org/10.1016/j.amc.2016.01.002 doi: 10.1016/j.amc.2016.01.002
[7]	Z. Tan, W. Wu, Global existence and decay estimate of solutions to magneto-micropolar fluid equations, J. Differ. Equations, 7 (2019), 4137–4169. https://doi.org/10.1016/j.jde.2018.09.027 doi: 10.1016/j.jde.2018.09.027
[8]	Y. Wang, K. Wang, Global well-posedness of 3D magneto-micropolar fluid equations with mixed partial viscosity, Nonlinear Anal-Real, 33 (2017), 348–362. https://doi.org/10.1016/j.nonrwa.2016.07.003 doi: 10.1016/j.nonrwa.2016.07.003
[9]	P. Zhang, M. Zhu, Global regularity of 3D nonhomogeneous incompressible magneto-micropolar system with the density-dependent viscosity, Comput. Math. Appl., 76 (2018), 2304–2314. https://doi.org/10.1016/j.camwa.2018.08.041 doi: 10.1016/j.camwa.2018.08.041
[10]	B. Yuan, Y. Qiao, Global regularity for the 2D magneto-micropolar equations with partial and fractional dissipation, Comput. Math. Appl., 76 (2018), 2345–2359. https://doi.org/10.1016/j.camwa.2018.08.029 doi: 10.1016/j.camwa.2018.08.029
[11]	K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Nonlinear Anal-Real, 35 (2015), 21932207. https://doi.org/10.3934/dcds.2015.35.2193 doi: 10.3934/dcds.2015.35.2193
[12]	H. Shang, C. Gu, Global regularity and decay estimates for 2D magneto-micropolar equations with partial dissipation, Z. Angew. Math. Phys., 70 (2019), 22, Art. https://doi.org/85.10.1007/s00033-019-1129-8
[13]	H. Shang, J. Zhao, Global regularity for 2D magneto-micropolar equations with only micro-rotational velocity dissipation and magnetic diffusion, Nonlinear Anal-Theor., 150 (2017), 194–209. https://doi.org/10.1016/j.na.2016.11.011 doi: 10.1016/j.na.2016.11.011
[14]	Li. Deng, H. Shang, Global well-posedness for n-dimensional magneto-micropolar equations with hyperdissipation, Appl. Math. Lett., 111 (2021), 106610. https://doi.org/10.1016/j.aml.2020.106610 doi: 10.1016/j.aml.2020.106610
[15]	H. Shang, C. Gu, Large time behavior for two-dimensional magneto-micropolar equations with only micro-rotational dissipation and magnetic diffusion, Appl. Math. Lett., 99 (2020), 105977. https://doi.org/10.1016/j.aml.2019.07.008 doi: 10.1016/j.aml.2019.07.008
[16]	Y. Liu, Global well-posedness to the Cauchy problem of 2D density-dependent micropolar equations with large initial data and vacuum, J. Math. Anal. Appl., 132 (2020), 124294. https://doi.org/10.1016/j.jmaa.2020.124294 doi: 10.1016/j.jmaa.2020.124294
[17]	Z. Ye, Global regularity results for the 2D Boussinesq equations and micropolar equations with partial dissipation, J. Differ. Equations, 268 (2020), 910–944. https://doi.org/10.1016/j.jde.2019.08.037 doi: 10.1016/j.jde.2019.08.037
[18]	B. Dong, J. Li, J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differ. Equations, 262 (2017), 3488–3523. https://doi.org/10.1016/j.jde.2016.11.029 doi: 10.1016/j.jde.2016.11.029
[19]	R. Guterres, W. Melo, J. Nunes, C. Perusato, On the large time decay of asymmetric flows in homogeneous Sobolev space, J. Math. Anal. Appl., 471 (2019), 88–101. https://doi.org/10.1016/j.jmaa.2018.10.065 doi: 10.1016/j.jmaa.2018.10.065
[20]	B. Dong, J. Wu, X. Xu, Z. Ye, Global regularity for the 2D micropolar equations with fractional dissipation, Discrete Contin. Dyn. Syst., 38 (2018), 41334162.
[21]	Q. Chen, C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differ. Equations, 252 (2012), 2698–2724. https://doi.org/10.1016/j.jde.2011.09.035 doi: 10.1016/j.jde.2011.09.035
[22]	W. Tan, B. Dong, Z. Chen, Regularity criterion for a critical fractional diffusion model of two-dimensional micropolar flows, J. Math. Anal. Appl., 470 (2019), 500–514. https://doi.org/10.1016/j.jmaa.2018.10.017 doi: 10.1016/j.jmaa.2018.10.017
[23]	M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635–664. https://doi.org/10.1016/0167-7136(83)90286-X doi: 10.1016/0167-7136(83)90286-X
[24]	C. Cao, J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803–1822. https://doi.org/10.1016/j.aim.2010.08.017 doi: 10.1016/j.aim.2010.08.017
[25]	N. Boardman, H. Lin, J. Wu, Stabilization of a background magnetic field on a 2 dimensional magnetohydrodynamic flow, SIAM J. Math. Anal., 52 (2020), 5001–5035. https://doi.org/10.1137/20M1324776 doi: 10.1137/20M1324776
[26]	B. Dong, Y. Jia, J. Li, J. Wu, Global regularity and time decay for the 2D magnetohydrodynamic equations with fractional dissipation and partial magnetic diffusion, J. Math. Fluid Mech., 20 (2018), 1541–1565. https://doi.org/10.1007/s00021-018-0376-3 doi: 10.1007/s00021-018-0376-3
[27]	H. Lin, R. Ji, J. Wu, L. Yan, Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation, J. Funct. Anal., 279 (2020), 108519. https://doi.org/10.1016/j.jfa.2020.108519 doi: 10.1016/j.jfa.2020.108519
[28]	S. Lai, J. Wu, J. Zhang, Stabilizing phenomenon for 2D anisotropic magnetohydrodynamic system near a background magnetic field, SIAM J. Math. Anal., 53 (2021), https://doi.org/6073-6093.10.1137/21M139791X
[29]	W. Feng, F. Hafeez, J. Wu, Influence of a background magnetic field on a 2D magnetohydrodynamic flow, Nonlinerity, 34 (2021), 2527–2562. https://doi.org/10.1088/1361-6544/abb928 doi: 10.1088/1361-6544/abb928
[30]	T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Providence, RI: American Mathematical Society, 2006.

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