# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022102
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## Global well-posedness to the nonhomogeneous magneto-micropolar fluid equations with large initial data and vacuum

 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  November 2021 Early access June 2022

Fund Project: This research was partially supported by National Natural Science Foundation of China (Nos. 11901474, 12071359) and Exceptional Young Talents Project of Chongqing Talent (No. cstc2021ycjh-bgzxm0153)

We investigate global well-posedness to nonhomogeneous magneto-micropolar fluid equations with zero density at infinity in $\mathbb{R}^2$. We show the global existence and uniqueness of strong solutions. It should be pointed out that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Our method relies crucially upon the duality principle of BMO space and Hardy space, a lemma of Coifman-Lions-Meyer-Semmes (Coifman et al. in J Math Pures Appl 72: 247–286, 1993), and cancelation properties of the system under consideration.

Citation: Xin Zhong. Global well-posedness to the nonhomogeneous magneto-micropolar fluid equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022102
##### References:
 [1] G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1. [2] J. L. Boldrini, M. A. Rojas-Medar and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl., 82 (2003), 1499-1525.  doi: 10.1016/j.matpur.2003.09.005. [3] P. Braz e Silva, F. W. Cruz, M. 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] R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. [5] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.  doi: 10.1016/j.jde.2004.07.002. [6] J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5 (2019), 37pp. doi: 10.1007/s40818-019-0064-5. [7] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. I. Incompressible Models, Oxford University Press, New York, 1996. [8] L. Liu and X. Zhong, Global existence and exponential decay of strong solutions for 2D nonhomogeneous micropolar fluids with density-dependent viscosity, J. Math. Phys., 62 (2021), 15pp. doi: 10.1063/5.0055689. [9] B. Lü, Z. 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. [10] G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5. [11] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13 (1959), 115-162. [12] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.  doi: 10.1515/9781400883929. [13] T. Tang and J. Sun, Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 6017-6026.  doi: 10.3934/dcdsb.2020377. [14] G. Wu and X. Zhong, Global strong solution and exponential decay of 3D nonhomogeneous asymmetric fluid equations with vacuum, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 1428-1444.  doi: 10.1007/s10473-021-0503-8. [15] X. Yang and X. Zhong, Global well-posedness and decay estimates to the 3D Cauchy problem of nonhomogeneous magneto-micropolar fluid equations with vacuum, J. Math. Phys., 63 (2022), 16pp. doi: 10.1063/5.0078216. [16] 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. [17] P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible magneto-micropolar system with the density-dependent viscosity, Comput. Math. Appl., 76 (2018), 2304-2314.  doi: 10.1016/j.camwa.2018.08.041. [18] P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1. [19] X. Zhong, Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum, to appear, Discrete Contin. Dyn. Syst. Ser. B. doi: 10.3934/dcdsb.2021296. [20] 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. [21] X. Zhong, Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density, Anal. Appl. (Singap.), 19 (2021), 245-273.  doi: 10.1142/S0219530519500167. [22] X. Zhong, Singularity formation to the nonhomogeneous magneto-micropolar fluid equations, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 6339-6357.  doi: 10.3934/dcdsb.2021021.

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##### References:
 [1] G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1. [2] J. L. Boldrini, M. A. Rojas-Medar and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl., 82 (2003), 1499-1525.  doi: 10.1016/j.matpur.2003.09.005. [3] P. Braz e Silva, F. W. Cruz, M. 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] R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. [5] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.  doi: 10.1016/j.jde.2004.07.002. [6] J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5 (2019), 37pp. doi: 10.1007/s40818-019-0064-5. [7] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. I. Incompressible Models, Oxford University Press, New York, 1996. [8] L. Liu and X. Zhong, Global existence and exponential decay of strong solutions for 2D nonhomogeneous micropolar fluids with density-dependent viscosity, J. Math. Phys., 62 (2021), 15pp. doi: 10.1063/5.0055689. [9] B. Lü, Z. 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. [10] G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5. [11] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13 (1959), 115-162. [12] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.  doi: 10.1515/9781400883929. [13] T. Tang and J. Sun, Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 6017-6026.  doi: 10.3934/dcdsb.2020377. [14] G. Wu and X. Zhong, Global strong solution and exponential decay of 3D nonhomogeneous asymmetric fluid equations with vacuum, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 1428-1444.  doi: 10.1007/s10473-021-0503-8. [15] X. Yang and X. Zhong, Global well-posedness and decay estimates to the 3D Cauchy problem of nonhomogeneous magneto-micropolar fluid equations with vacuum, J. Math. Phys., 63 (2022), 16pp. doi: 10.1063/5.0078216. [16] 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. [17] P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible magneto-micropolar system with the density-dependent viscosity, Comput. Math. Appl., 76 (2018), 2304-2314.  doi: 10.1016/j.camwa.2018.08.041. [18] P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1. [19] X. Zhong, Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum, to appear, Discrete Contin. Dyn. Syst. Ser. B. doi: 10.3934/dcdsb.2021296. [20] 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. [21] X. Zhong, Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density, Anal. Appl. (Singap.), 19 (2021), 245-273.  doi: 10.1142/S0219530519500167. [22] X. Zhong, Singularity formation to the nonhomogeneous magneto-micropolar fluid equations, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 6339-6357.  doi: 10.3934/dcdsb.2021021.
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