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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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

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

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

[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.

[26]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

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

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

[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.

[26]

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

[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.

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