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Global large smooth solutions for 3-D Hall-magnetohydrodynamics
Changsha University of Science and Technology, School of Mathematics and Statistics, 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 [
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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