doi: 10.3934/dcdsb.2022061
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Global solvability to the 3D incompressible magneto-micropolar system with vacuum

1. 

School of Mathematics, Jilin University, Changchun 130012, China

2. 

College of Mathematics, Changchun Normal University, Changchun 130032, China

*Corresponding author: Yang Liu

Received  November 2021 Revised  January 2022 Early access March 2022

Fund Project: The first author is supported by NSF of China (11901288), Postdoctoral Science Foundation of China (2021M691219), Scientific Research Foundation of Jilin Provincial Education Department (JJKH20210 873KJ), and Natural Science Foundation of Changchun Normal University

This paper deals with the Cauchy problem of 3D innhomogeneous incompressible magneto-micropolar system. We prove the global existence of strong solutions to this system, with initial data being of small norm but allowed to have vacuum and large oscillations. More precisely, we only require that the initial data
$ (\rho_0, u_0, w_0, b_0) $
satisfying
$ \begin{align*} &\Big(\|\sqrt{\rho_0}u_0\|_{L^2}^2+\|\sqrt{\rho_0}w_0\|_{L^2}^2+\|b_0\|_{L^2}^2\Big)\times\Big(\mu_1\|\nabla u_0\|_{L^2}^2 +\mu_2\|\nabla w_0\|_{L^2}^2\nonumber\\ &\quad+(\mu_2+\lambda)\|{\rm div}w_0\|_{L^2}^2+\eta\|\nabla b_0\|_{L^2}^2 +\xi\|2w_0-\nabla\times u_0\|_{L^2}^2\Big) \end{align*} $
is suitably small, which extends the corresponding Cruz and Novais's result (Appl. Anal., 2020[9]) to the inhomogeneous case, and Ye's result (Discrete Contin. Dyn. Syst. B, 2019[17]) to the 3D Cauchy problem of the inhomogeneous micropolar equations with magnetic field. Furthermore, we also established the large time behavior of strong solutions.
Citation: Yang Liu, Nan Zhou, Renying Guo. Global solvability to the 3D incompressible magneto-micropolar system with vacuum. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022061
References:
[1]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476.  doi: 10.1017/S0308210506001181.

[2]

P. Braz e SilvaF. Cruz and M. Rojas-Medar, Vanishing viscosity for non-homogeneous asymmetric fluids in $\Bbb{R}^3$: The $L^2$ case, J. Math. Anal. Appl., 420 (2014), 207-221.  doi: 10.1016/j.jmaa.2014.05.060.

[3]

P. Braz e SilvaF. W. CruzM. Loayza and M. A. Rojas-Medar, Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach, J. Differential Equations, 269 (2020), 1319-1348.  doi: 10.1016/j.jde.2020.01.001.

[4]

P. Braz e SilvaE. Fernádez-Cara and M. Rojas-Medar, Vanishing viscosity for non-homogeneous asymmetric fluids in ${\Bbb R}^3$, J. Math. Anal. Appl., 332 (2007), 833-845.  doi: 10.1016/j.jmaa.2006.10.066.

[5]

P. Braz e Silva and E. Santos, Global weak solutions for asymmetric incompressible fluids with variable density, C. R. Math. Acad. Sci. Paris, 346 (2008), 575-578.  doi: 10.1016/j.crma.2008.03.008.

[6]

P. Braz e Silva and E. Santos, Global weak solutions for variable density asymmetric incompressible fluids, J. Math. Anal. Appl., 387 (2012), 953-969.  doi: 10.1016/j.jmaa.2011.10.015.

[7]

F. ChenB. Guo and X. Zhai, Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density, Kinet. Relat. Models, 12 (2019), 37-58.  doi: 10.3934/krm.2019002.

[8]

Q. ChenZ. Tan and Y. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.  doi: 10.1002/mma.1338.

[9]

F. W. Cruz and M. M. Novais, On the strong solutions of the 3D magneto-micropolar equations, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1791831.

[10]

H. Gong and J. Li, Global existence of strong solutions to incompressible MHD, Commun. Pure Appl. Anal., 13 (2014), 1553-1561.  doi: 10.3934/cpaa.2014.13.1553.

[11]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029.

[12]

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., 491 (2020), 124294, 15 pp. doi: 10.1016/j.jmaa.2020.124294.

[13]

B. LvZ. Xu and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.  doi: 10.1016/j.matpur.2016.10.009.

[14] A. Novotny and I. Straŝkraba, Introduction to The Mathematical Theory of Compressible Flow, Oxford University Press, 2004. 
[15]

S. Song, On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum, Z. Angew. Math. Phys., 69 (2018), 27 pp. doi: 10.1007/s00033-018-0915-z.

[16]

G. Wu and X. Zhong, Global strong solution and exponential decay of 3D nonhomogeneous asymmetric fluid equations with vacuum, Acta Math. Sci., 41 (2021), 1428-1444.  doi: 10.1007/s10473-021-0503-8.

[17]

Z. Ye, Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6725-6743.  doi: 10.3934/dcdsb.2019164.

[18]

J. Zhang and J. Zhao, Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics, Commun. Math. Sci., 8 (2010), 835-850.  doi: 10.4310/CMS.2010.v8.n4.a2.

[19]

P. Zhang and H. Yu, Global regularity to the 3D incompressible MHD equations, J. Math. Anal. Appl., 432 (2015), 613-631.  doi: 10.1016/j.jmaa.2015.07.007.

[20]

P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incomressible micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1.

[21]

X. Zhong, Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 3563-3578.  doi: 10.3934/dcdsb.2020246.

[22]

X. Zhong, Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum, Commun. Pure Appl. Anal., 21 (2022), 493-515.  doi: 10.3934/cpaa.2021185.

[23]

X. Zhong, Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum, Discrete Contin. Dyn. Syst. Ser. B, early access, 2022. doi: 10.3934/dcds-b.2021296.

show all references

References:
[1]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476.  doi: 10.1017/S0308210506001181.

[2]

P. Braz e SilvaF. Cruz and M. Rojas-Medar, Vanishing viscosity for non-homogeneous asymmetric fluids in $\Bbb{R}^3$: The $L^2$ case, J. Math. Anal. Appl., 420 (2014), 207-221.  doi: 10.1016/j.jmaa.2014.05.060.

[3]

P. Braz e SilvaF. W. CruzM. Loayza and M. A. Rojas-Medar, Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach, J. Differential Equations, 269 (2020), 1319-1348.  doi: 10.1016/j.jde.2020.01.001.

[4]

P. Braz e SilvaE. Fernádez-Cara and M. Rojas-Medar, Vanishing viscosity for non-homogeneous asymmetric fluids in ${\Bbb R}^3$, J. Math. Anal. Appl., 332 (2007), 833-845.  doi: 10.1016/j.jmaa.2006.10.066.

[5]

P. Braz e Silva and E. Santos, Global weak solutions for asymmetric incompressible fluids with variable density, C. R. Math. Acad. Sci. Paris, 346 (2008), 575-578.  doi: 10.1016/j.crma.2008.03.008.

[6]

P. Braz e Silva and E. Santos, Global weak solutions for variable density asymmetric incompressible fluids, J. Math. Anal. Appl., 387 (2012), 953-969.  doi: 10.1016/j.jmaa.2011.10.015.

[7]

F. ChenB. Guo and X. Zhai, Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density, Kinet. Relat. Models, 12 (2019), 37-58.  doi: 10.3934/krm.2019002.

[8]

Q. ChenZ. Tan and Y. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.  doi: 10.1002/mma.1338.

[9]

F. W. Cruz and M. M. Novais, On the strong solutions of the 3D magneto-micropolar equations, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1791831.

[10]

H. Gong and J. Li, Global existence of strong solutions to incompressible MHD, Commun. Pure Appl. Anal., 13 (2014), 1553-1561.  doi: 10.3934/cpaa.2014.13.1553.

[11]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029.

[12]

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., 491 (2020), 124294, 15 pp. doi: 10.1016/j.jmaa.2020.124294.

[13]

B. LvZ. Xu and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.  doi: 10.1016/j.matpur.2016.10.009.

[14] A. Novotny and I. Straŝkraba, Introduction to The Mathematical Theory of Compressible Flow, Oxford University Press, 2004. 
[15]

S. Song, On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum, Z. Angew. Math. Phys., 69 (2018), 27 pp. doi: 10.1007/s00033-018-0915-z.

[16]

G. Wu and X. Zhong, Global strong solution and exponential decay of 3D nonhomogeneous asymmetric fluid equations with vacuum, Acta Math. Sci., 41 (2021), 1428-1444.  doi: 10.1007/s10473-021-0503-8.

[17]

Z. Ye, Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6725-6743.  doi: 10.3934/dcdsb.2019164.

[18]

J. Zhang and J. Zhao, Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics, Commun. Math. Sci., 8 (2010), 835-850.  doi: 10.4310/CMS.2010.v8.n4.a2.

[19]

P. Zhang and H. Yu, Global regularity to the 3D incompressible MHD equations, J. Math. Anal. Appl., 432 (2015), 613-631.  doi: 10.1016/j.jmaa.2015.07.007.

[20]

P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incomressible micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1.

[21]

X. Zhong, Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 3563-3578.  doi: 10.3934/dcdsb.2020246.

[22]

X. Zhong, Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum, Commun. Pure Appl. Anal., 21 (2022), 493-515.  doi: 10.3934/cpaa.2021185.

[23]

X. Zhong, Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum, Discrete Contin. Dyn. Syst. Ser. B, early access, 2022. doi: 10.3934/dcds-b.2021296.

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