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November  2019, 39(11): 6669-6682. doi: 10.3934/dcds.2019290

Global large smooth solutions for 3-D Hall-magnetohydrodynamics

Huali Zhang ,

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.

Keywords: Incompressible Hall-MHD equations, large data, global smooth solution.
Mathematics Subject Classification: Primary: 35A01, 35A02, 35A09; Secondary: 35E15, 35Q35.
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. Acheritogaray, P. Degond, A. 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. Chae, P. 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. Lei, F. 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. Lin, H. 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. Mininni, D. 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. Ren, J. H. Wu, Z. 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. Acheritogaray, P. Degond, A. 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. Chae, P. 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. Lei, F. 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. Lin, H. 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. Mininni, D. 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. Ren, J. H. Wu, Z. 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

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