• Previous Article
    Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting
  • DCDS Home
  • This Issue
  • Next Article
    Scattering of radial data in the focusing NLS and generalized Hartree equations
November  2019, 39(11): 6669-6682. doi: 10.3934/dcds.2019290

Global large smooth solutions for 3-D Hall-magnetohydrodynamics

Changsha University of Science and Technology, School of Mathematics and Statistics, Changsha 410114, China

* Corresponding author: Huali Zhang

Received  March 2019 Published  August 2019

Fund Project: The first author is supported by Education Department of Hunan Province, general Program(grant No.17C0039), and Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, Changsha University of Science and Technology, Changsha; 410114, China

In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data is constructed, which can be arbitrarily large in $ H^3(\mathbb{R}^3) $. Our result may also be considered as the extension of work of Lei-Lin-Zhou [15] from the second-order semilinear equations to the second-order quasilinear equations, because the Hall term elevates the Hall-MHD system to the quasilinear level.

Citation: Huali Zhang. Global large smooth solutions for 3-D Hall-magnetohydrodynamics. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6669-6682. doi: 10.3934/dcds.2019290
References:
[1]

M. AcheritogarayP. DegondA. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system, Kinet. Relat. Models, 4 (2011), 901-918. doi: 10.3934/krm.2011.4.901. Google Scholar

[2]

S. A. Balbus and C. Terquem, Linear analysis of the Hall effect in protostellar disks, The Astrophysical Journal, 552 (2001), 235-247. doi: 10.1086/320452. Google Scholar

[3]

D. ChaeP. Degond and J. G. Liu, Well-posedness for Hallmagnetohydrodynamics, Ann. I. H. Poincaré, 31 (2014), 555-565. doi: 10.1016/j.anihpc.2013.04.006. Google Scholar

[4]

D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall- magneto-hydrodynamics, J. Differential Equations, 256 (2014), 3835-3858. doi: 10.1016/j.jde.2014.03.003. Google Scholar

[5]

D. Chae and M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), 3971-3982. doi: 10.1016/j.jde.2013.07.059. Google Scholar

[6]

D. Chae, R. Wan and J. Wu, Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion, J. Math. Fluid Mech. 17 (2015), 627–638. doi: 10.1007/s00021-015-0222-9. Google Scholar

[7]

D. Chae and S. Weng, Singularity formation for the incompressible Hall-MHD equations without resistivity, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 33 (2016), 1009-1022. doi: 10.1016/j.anihpc.2015.03.002. Google Scholar

[8]

D. Chae and J. Wolf, On partial regularity for the 3D non-stationary Hall magnetohydrodynamics equations on the plane, SIAM J. Math. Anal., 48 (2016), 443-469. doi: 10.1137/15M1012037. Google Scholar

[9]

J. Y. Chemin and I. Gallagher, Well-posedness and stability results for the Navier-Stokes equa tions in R3, Ann. Inst. H. H. Poincaré Anal. Non Lineaire, 26 (2009), 599-624. doi: 10.1016/j.anihpc.2007.05.008. Google Scholar

[10]

P. Constantin and A. Majda, The Beltrami spectrum for incompressible fluid flows, Commun. Math. Phys., 115 (1988), 435-456. doi: 10.1007/BF01218019. Google Scholar

[11]

M. M. Dai, Local well-posedness for the Hall-MHD system in optimal Sobolev spaces, preprint, arXiv: 1803.09556.Google Scholar

[12] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511626333. Google Scholar
[13]

T. G. Forbes, Magnetic reconnection in solar flares, Geophysical and Astrophysical Fluid Dynamics, 62 (1991), 15-36. doi: 10.1080/03091929108229123. Google Scholar

[14]

H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Physica D., 208 (2005), 59-72. doi: 10.1016/j.physd.2005.06.003. Google Scholar

[15]

Z. LeiF. H. Lin and Y. Zhou, Structure of helicity and global solutions of incompressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 218 (2015), 1417-1430. doi: 10.1007/s00205-015-0884-8. Google Scholar

[16]

M. J. Lighthill, Studies on magnetohydrodynamic waves and other anisotropic wave motions. Philos,, Trans. R. Soc. Lond., Ser., 252 (1960), 397-430. doi: 10.1098/rsta.1960.0010. Google Scholar

[17]

F. H. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506. Google Scholar

[18]

Y. R. LinH. L. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112. doi: 10.1016/j.jde.2016.03.002. Google Scholar

[19]

P. D. MininniD. O. Gómez and S. M. Mahajan, Dynamo action in magnetohydrodynamics and Hall magnetohydrodynamics, The Astrophysics Journal, 587 (2003), 472-481. doi: 10.1086/368181. Google Scholar

[20]

X. X. RenJ. H. WuZ. Y. Xiang and Z. F. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541. doi: 10.1016/j.jfa.2014.04.020. Google Scholar

[21]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506. Google Scholar

[22]

D. A. Shalybkov and V. A. Urpin, The Hall effect and the decay of magnetic fields, Astron. Astrophys., 321 (1997), 685-690. Google Scholar

[23] E. M. Stein, Singular Integrals and Differentialbility Properties of Functions, Princeton University Press, Princeton, 1970. Google Scholar
[24]

J. B. Taylor, Relaxation of toroidal plasma and generation of reverse magnetic fields, Phy. Rev. Letter, 33 (1974), 1138-1141. Google Scholar

[25]

M. Wardle, Star formation and the Hall effect, Magnetic Fields and Star Formation, (2004), 231–237. doi: 10.1007/978-94-017-0491-5_24. Google Scholar

[26]

K. Yamazaki and M. T. Moha, Well-posedness of Hall-magnetohydrodynamics system forced by Lévy noise, Stoch. PDE: Anal. Comp., (2018), 1–48.Google Scholar

[27]

Y. Zhou and Y. Zhu, A class of large solutions to the 3D incompressible MHD and Euler equations with damping, Acta Math. Sinica English Series, 34 (2018), 63-78. doi: 10.1007/s10114-016-6271-z. Google Scholar

show all references

References:
[1]

M. AcheritogarayP. DegondA. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system, Kinet. Relat. Models, 4 (2011), 901-918. doi: 10.3934/krm.2011.4.901. Google Scholar

[2]

S. A. Balbus and C. Terquem, Linear analysis of the Hall effect in protostellar disks, The Astrophysical Journal, 552 (2001), 235-247. doi: 10.1086/320452. Google Scholar

[3]

D. ChaeP. Degond and J. G. Liu, Well-posedness for Hallmagnetohydrodynamics, Ann. I. H. Poincaré, 31 (2014), 555-565. doi: 10.1016/j.anihpc.2013.04.006. Google Scholar

[4]

D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall- magneto-hydrodynamics, J. Differential Equations, 256 (2014), 3835-3858. doi: 10.1016/j.jde.2014.03.003. Google Scholar

[5]

D. Chae and M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), 3971-3982. doi: 10.1016/j.jde.2013.07.059. Google Scholar

[6]

D. Chae, R. Wan and J. Wu, Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion, J. Math. Fluid Mech. 17 (2015), 627–638. doi: 10.1007/s00021-015-0222-9. Google Scholar

[7]

D. Chae and S. Weng, Singularity formation for the incompressible Hall-MHD equations without resistivity, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 33 (2016), 1009-1022. doi: 10.1016/j.anihpc.2015.03.002. Google Scholar

[8]

D. Chae and J. Wolf, On partial regularity for the 3D non-stationary Hall magnetohydrodynamics equations on the plane, SIAM J. Math. Anal., 48 (2016), 443-469. doi: 10.1137/15M1012037. Google Scholar

[9]

J. Y. Chemin and I. Gallagher, Well-posedness and stability results for the Navier-Stokes equa tions in R3, Ann. Inst. H. H. Poincaré Anal. Non Lineaire, 26 (2009), 599-624. doi: 10.1016/j.anihpc.2007.05.008. Google Scholar

[10]

P. Constantin and A. Majda, The Beltrami spectrum for incompressible fluid flows, Commun. Math. Phys., 115 (1988), 435-456. doi: 10.1007/BF01218019. Google Scholar

[11]

M. M. Dai, Local well-posedness for the Hall-MHD system in optimal Sobolev spaces, preprint, arXiv: 1803.09556.Google Scholar

[12] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511626333. Google Scholar
[13]

T. G. Forbes, Magnetic reconnection in solar flares, Geophysical and Astrophysical Fluid Dynamics, 62 (1991), 15-36. doi: 10.1080/03091929108229123. Google Scholar

[14]

H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Physica D., 208 (2005), 59-72. doi: 10.1016/j.physd.2005.06.003. Google Scholar

[15]

Z. LeiF. H. Lin and Y. Zhou, Structure of helicity and global solutions of incompressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 218 (2015), 1417-1430. doi: 10.1007/s00205-015-0884-8. Google Scholar

[16]

M. J. Lighthill, Studies on magnetohydrodynamic waves and other anisotropic wave motions. Philos,, Trans. R. Soc. Lond., Ser., 252 (1960), 397-430. doi: 10.1098/rsta.1960.0010. Google Scholar

[17]

F. H. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506. Google Scholar

[18]

Y. R. LinH. L. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112. doi: 10.1016/j.jde.2016.03.002. Google Scholar

[19]

P. D. MininniD. O. Gómez and S. M. Mahajan, Dynamo action in magnetohydrodynamics and Hall magnetohydrodynamics, The Astrophysics Journal, 587 (2003), 472-481. doi: 10.1086/368181. Google Scholar

[20]

X. X. RenJ. H. WuZ. Y. Xiang and Z. F. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541. doi: 10.1016/j.jfa.2014.04.020. Google Scholar

[21]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506. Google Scholar

[22]

D. A. Shalybkov and V. A. Urpin, The Hall effect and the decay of magnetic fields, Astron. Astrophys., 321 (1997), 685-690. Google Scholar

[23] E. M. Stein, Singular Integrals and Differentialbility Properties of Functions, Princeton University Press, Princeton, 1970. Google Scholar
[24]

J. B. Taylor, Relaxation of toroidal plasma and generation of reverse magnetic fields, Phy. Rev. Letter, 33 (1974), 1138-1141. Google Scholar

[25]

M. Wardle, Star formation and the Hall effect, Magnetic Fields and Star Formation, (2004), 231–237. doi: 10.1007/978-94-017-0491-5_24. Google Scholar

[26]

K. Yamazaki and M. T. Moha, Well-posedness of Hall-magnetohydrodynamics system forced by Lévy noise, Stoch. PDE: Anal. Comp., (2018), 1–48.Google Scholar

[27]

Y. Zhou and Y. Zhu, A class of large solutions to the 3D incompressible MHD and Euler equations with damping, Acta Math. Sinica English Series, 34 (2018), 63-78. doi: 10.1007/s10114-016-6271-z. Google Scholar

[1]

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

[2]

Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077

[3]

Ning Duan, Yasuhide Fukumoto, Xiaopeng Zhao. Asymptotic behavior of solutions to incompressible electron inertial Hall-MHD system in $ \mathbb{R}^3 $. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3035-3057. doi: 10.3934/cpaa.2019136

[4]

Jishan Fan, Fucai Li, Gen Nakamura. Low Mach number limit of the full compressible Hall-MHD system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1731-1740. doi: 10.3934/cpaa.2017084

[5]

Zhong Tan, Huaqiao Wang, Yucong Wang. Time-splitting methods to solve the Hall-MHD systems with Lévy noises. Kinetic & Related Models, 2019, 12 (1) : 243-267. doi: 10.3934/krm.2019011

[6]

Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521

[7]

Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic & Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002

[8]

Ming He, Jianwen Zhang. Global cylindrical solution to the compressible MHD equations in an exterior domain. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1841-1865. doi: 10.3934/cpaa.2009.8.1841

[9]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553

[10]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337

[11]

Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865

[12]

Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873

[13]

Hua Qiu, Shaomei Fang. A BKM's criterion of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2014, 13 (2) : 823-833. doi: 10.3934/cpaa.2014.13.823

[14]

Baoquan Yuan. Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2211-2219. doi: 10.3934/dcds.2013.33.2211

[15]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

[16]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209

[17]

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

[18]

Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122

[19]

Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026

[20]

Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016

2018 Impact Factor: 1.143

Article outline

[Back to Top]