doi: 10.3934/nhm.2022021
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On the local and global existence of the Hall equations with fractional Laplacian and related equations

Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Republic of Korea

*Corresponding author: Hantaek Bae

Received  November 2021 Revised  February 2022 Early access May 2022

Fund Project: The first author is supported by NRF-2018R1D1A1B07049015

In this paper, we deal with the Hall equations with fractional Laplacian
$ B_{t}+{\rm{curl}} \left(({\rm{curl}} \;B)\times B\right)+\Lambda B = 0. $
We begin to prove the existence of unique global in time solutions with sufficiently small initial data in
$ H^{k} $
,
$ k>\frac{5}{2} $
. By correcting
$ \Lambda B $
logarithmically, we then show the existence of unique local in time solutions. We also deal with the two dimensional systems closely related to the
$ 2\frac{1}{2} $
dimensional version of the above Hall equations. In this case, we show the existence of unique local and global in time solutions depending on whether the damping term is present or not.
Citation: Hantaek Bae. On the local and global existence of the Hall equations with fractional Laplacian and related equations. Networks and Heterogeneous Media, doi: 10.3934/nhm.2022021
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.

[2]

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

[3]

A. J. Brizard, Comment on Exact solutions and singularities of an $X$-point collapse in Hall magnetohydrodynamics [J. Math. Phys. 59, 061509 (2018)], J. Math. Phys., 60 (2019), 024101, 6 pp. doi: 10.1063/1.5090490.

[4]

D. ChaeP. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565.  doi: 10.1016/j.anihpc.2013.04.006.

[5]

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

[6]

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.

[7]

D. ChaeR. 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.

[8]

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

[9]

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

[10]

D. Chae and J. Wolf, Regularity of the 3D stationary hall magnetohydrodynamic equations on the plane, Comm. Math. Phys., 354 (2017), 213-230.  doi: 10.1007/s00220-017-2908-8.

[11]

F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.

[12]

M. Dai and H. Liu, Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion, J. Differential Equations, 266 (2019), 7658-7677.  doi: 10.1016/j.jde.2018.12.008.

[13]

J. FanY. FukumotoG. Nakamura and Y. Zhou, Regularity criteria for the incompressible Hall-MHD system, ZAMM Z. Angew. Math. Mech., 95 (2015), 1156-1160.  doi: 10.1002/zamm.201400102.

[14]

J. FanS. Huang and G. Nakamura, Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations, Appl. Math. Lett., 26 (2013), 963-967.  doi: 10.1016/j.aml.2013.04.008.

[15]

J. FanX. JiaG. Nakamura and Y. Zhou, On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects, Z. Angew. Math. Phys., 66 (2015), 1695-1706.  doi: 10.1007/s00033-015-0499-9.

[16]

J. FanF. Li and G. Nakamura, Regularity criteria for the incompressible Hall-magnetohydrodynamic equations, Nonlinear Anal., 109 (2014), 173-179.  doi: 10.1016/j.na.2014.07.003.

[17]

T. G. Forbes, Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 62 (1991), 15-36.  doi: 10.1080/03091929108229123.

[18]

W. J. Han, H. J. Hwang and B. S. Moon, On the well-posedness of the Hall-magnetohydrodynamics with the ion-slip effect, J. Math. Fluid Mech., 21 (2019), Paper No. 47, 28 pp. doi: 10.1007/s00021-019-0455-0.

[19]

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

[20]

A. Z. Janda, Exact solutions and singularities of an $X$-point collapse in Hall magnetohydrodynamics, J. Math. Phys., 59 (2018), 061509, 11 pp. doi: 10.1063/1.5026876.

[21]

A. Z. Janda, Response to Comment on Exact solutions and singularities of an X-point collapse in Hall magnetohydrodynamics [J. Math. Phys. 60, 024101 (2019)], J. Math. Phys., 60 (2019), 024102, 3 pp. doi: 10.1063/1.5078768.

[22]

T. Kato and G. Ponce, Commutator cstimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[23]

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

[24]

Y. E. Litavinenko and L. C. McMahon, Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics, Appl. Math. Lett., 45 (2015), 76-80.  doi: 10.1016/j.aml.2015.01.012.

[25]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. xii+545 pp.

[26]

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

[27]

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

[28]

M. A. ShayJ. F. DrakeR. E. Denton and D. Biskamp, Structure of the dissipation region during collisionless magnetic reconnection, Journal of Geophysical Research, 103 (1998), 9165-9176.  doi: 10.1029/97JA03528.

[29]

R. Wan and Y. Zhou, On global existence, energy decay and blow-up criteria for the Hall-MHD system, J. Differential Equations, 259 (2015), 5982-6008.  doi: 10.1016/j.jde.2015.07.013.

[30]

R. Wan and Y. Zhou, Low regularity well-posedness for the 3D generalized Hall-MHD system, Acta Appl. Math., 147 (2017), 95-111.  doi: 10.1007/s10440-016-0070-5.

[31]

R. Wan and Y. Zhou, Global well-posedness for the 3D incompressible Hall-magnetohydrodynamic equations with Fujita-Kato type initial data, J. Math. Fluid Mech., 21 (2019), Paper No. 5, 16 pp. doi: 10.1007/s00021-019-0410-0.

[32]

R. Wan and Y. Zhou, Global well-posedness, BKM blow-up criteria and zero $h$ limit for the 3D incompressible Hall-MHD equations, J. Differential Equations, 267 (2019), 3724-3747.  doi: 10.1016/j.jde.2019.04.020.

[33]

M. Wardle, Star formation and the Hall effect, Astrophys. Space Sci., 292 (2004), 317-323.  doi: 10.1023/B:ASTR.0000045033.80068.1f.

[34]

S. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differential Equations, 260 (2016), 6504-6524.  doi: 10.1016/j.jde.2016.01.003.

[35]

S. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270 (2016), 2168-2187.  doi: 10.1016/j.jfa.2016.01.021.

[36]

K. Yamazaki, Irreducibility of the three, and two and a half dimensional Hall-magnetohydrodynamics system, Phys. D, 401 (2020), 132199, 21 pp. doi: 10.1016/j.physd.2019.132199.

[37]

H. Zhang and K. Zhao, On 3D Hall-MHD equations with fractional Laplacians: Global well-posedness, J. Math. Fluid Mech., 23 (2021), Paper No. 82, 25 pp. doi: 10.1007/s00021-021-00605-y.

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.

[2]

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

[3]

A. J. Brizard, Comment on Exact solutions and singularities of an $X$-point collapse in Hall magnetohydrodynamics [J. Math. Phys. 59, 061509 (2018)], J. Math. Phys., 60 (2019), 024101, 6 pp. doi: 10.1063/1.5090490.

[4]

D. ChaeP. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565.  doi: 10.1016/j.anihpc.2013.04.006.

[5]

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

[6]

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.

[7]

D. ChaeR. 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.

[8]

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

[9]

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

[10]

D. Chae and J. Wolf, Regularity of the 3D stationary hall magnetohydrodynamic equations on the plane, Comm. Math. Phys., 354 (2017), 213-230.  doi: 10.1007/s00220-017-2908-8.

[11]

F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.

[12]

M. Dai and H. Liu, Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion, J. Differential Equations, 266 (2019), 7658-7677.  doi: 10.1016/j.jde.2018.12.008.

[13]

J. FanY. FukumotoG. Nakamura and Y. Zhou, Regularity criteria for the incompressible Hall-MHD system, ZAMM Z. Angew. Math. Mech., 95 (2015), 1156-1160.  doi: 10.1002/zamm.201400102.

[14]

J. FanS. Huang and G. Nakamura, Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations, Appl. Math. Lett., 26 (2013), 963-967.  doi: 10.1016/j.aml.2013.04.008.

[15]

J. FanX. JiaG. Nakamura and Y. Zhou, On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects, Z. Angew. Math. Phys., 66 (2015), 1695-1706.  doi: 10.1007/s00033-015-0499-9.

[16]

J. FanF. Li and G. Nakamura, Regularity criteria for the incompressible Hall-magnetohydrodynamic equations, Nonlinear Anal., 109 (2014), 173-179.  doi: 10.1016/j.na.2014.07.003.

[17]

T. G. Forbes, Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 62 (1991), 15-36.  doi: 10.1080/03091929108229123.

[18]

W. J. Han, H. J. Hwang and B. S. Moon, On the well-posedness of the Hall-magnetohydrodynamics with the ion-slip effect, J. Math. Fluid Mech., 21 (2019), Paper No. 47, 28 pp. doi: 10.1007/s00021-019-0455-0.

[19]

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

[20]

A. Z. Janda, Exact solutions and singularities of an $X$-point collapse in Hall magnetohydrodynamics, J. Math. Phys., 59 (2018), 061509, 11 pp. doi: 10.1063/1.5026876.

[21]

A. Z. Janda, Response to Comment on Exact solutions and singularities of an X-point collapse in Hall magnetohydrodynamics [J. Math. Phys. 60, 024101 (2019)], J. Math. Phys., 60 (2019), 024102, 3 pp. doi: 10.1063/1.5078768.

[22]

T. Kato and G. Ponce, Commutator cstimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[23]

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

[24]

Y. E. Litavinenko and L. C. McMahon, Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics, Appl. Math. Lett., 45 (2015), 76-80.  doi: 10.1016/j.aml.2015.01.012.

[25]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. xii+545 pp.

[26]

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

[27]

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

[28]

M. A. ShayJ. F. DrakeR. E. Denton and D. Biskamp, Structure of the dissipation region during collisionless magnetic reconnection, Journal of Geophysical Research, 103 (1998), 9165-9176.  doi: 10.1029/97JA03528.

[29]

R. Wan and Y. Zhou, On global existence, energy decay and blow-up criteria for the Hall-MHD system, J. Differential Equations, 259 (2015), 5982-6008.  doi: 10.1016/j.jde.2015.07.013.

[30]

R. Wan and Y. Zhou, Low regularity well-posedness for the 3D generalized Hall-MHD system, Acta Appl. Math., 147 (2017), 95-111.  doi: 10.1007/s10440-016-0070-5.

[31]

R. Wan and Y. Zhou, Global well-posedness for the 3D incompressible Hall-magnetohydrodynamic equations with Fujita-Kato type initial data, J. Math. Fluid Mech., 21 (2019), Paper No. 5, 16 pp. doi: 10.1007/s00021-019-0410-0.

[32]

R. Wan and Y. Zhou, Global well-posedness, BKM blow-up criteria and zero $h$ limit for the 3D incompressible Hall-MHD equations, J. Differential Equations, 267 (2019), 3724-3747.  doi: 10.1016/j.jde.2019.04.020.

[33]

M. Wardle, Star formation and the Hall effect, Astrophys. Space Sci., 292 (2004), 317-323.  doi: 10.1023/B:ASTR.0000045033.80068.1f.

[34]

S. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differential Equations, 260 (2016), 6504-6524.  doi: 10.1016/j.jde.2016.01.003.

[35]

S. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270 (2016), 2168-2187.  doi: 10.1016/j.jfa.2016.01.021.

[36]

K. Yamazaki, Irreducibility of the three, and two and a half dimensional Hall-magnetohydrodynamics system, Phys. D, 401 (2020), 132199, 21 pp. doi: 10.1016/j.physd.2019.132199.

[37]

H. Zhang and K. Zhao, On 3D Hall-MHD equations with fractional Laplacians: Global well-posedness, J. Math. Fluid Mech., 23 (2021), Paper No. 82, 25 pp. doi: 10.1007/s00021-021-00605-y.

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