doi: 10.3934/dcdsb.2022102
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. BoldriniM. 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 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]

R. CoifmanP.-L. LionsY. 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. LüZ. 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.

show all references

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. BoldriniM. 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 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]

R. CoifmanP.-L. LionsY. 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. LüZ. 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.

[1]

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

[2]

Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292

[3]

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

[4]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6017-6026. doi: 10.3934/dcdsb.2020377

[5]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6339-6357. doi: 10.3934/dcdsb.2021021

[6]

Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4201-4211. doi: 10.3934/dcdss.2021114

[7]

Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048

[8]

Hong Chen, Xin Zhong. Local well-posedness to the 2D Cauchy problem of non-isothermal nonhomogeneous nematic liquid crystal flows with vacuum at infinity. Communications on Pure and Applied Analysis, 2022, 21 (9) : 3141-3169. doi: 10.3934/cpaa.2022093

[9]

Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations and Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019

[10]

Hiroshi Inoue, Kei Matsuura, Mitsuharu Ôtani. Strong solutions of magneto-micropolar fluid equation. Conference Publications, 2003, 2003 (Special) : 439-448. doi: 10.3934/proc.2003.2003.439

[11]

Yang Liu, Nan Zhou, Renying Guo. Global solvability to the 3D incompressible magneto-micropolar system with vacuum. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022061

[12]

Kazuo Yamazaki. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2193-2207. doi: 10.3934/dcds.2015.35.2193

[13]

Jinbo Geng, Xiaochun Chen, Sadek Gala. On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space. Communications on Pure and Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583

[14]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181

[15]

Belkacem Said-Houari. Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation with arbitrarily large higher-order Sobolev norms. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4615-4635. doi: 10.3934/dcds.2022066

[16]

Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115

[17]

Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945

[18]

Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227

[19]

Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic and Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041

[20]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083

2021 Impact Factor: 1.497

Article outline

[Back to Top]