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September  2017, 16(5): 1731-1740. doi: 10.3934/cpaa.2017084

Low Mach number limit of the full compressible Hall-MHD system

Jishan Fan 1, , Fucai Li 2,, and  Gen Nakamura 3,

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China
3. 

Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

F. Li is the corresponding author

Received  October 2016 Revised  April 2017 Published  May 2017

In this paper we study the low Mach number limit of the full compressible Hall-magnetohydrodynamic (Hall-MHD) system in $\mathbb{T}^3$. We prove that, as the Mach number tends to zero, the strong solution of the full compressible Hall-MHD system converges to that of the incompressible Hall-MHD system.

Keywords: Full compressible Hall-MHD system, incompressible Hall-MHD system, low Mach number limit.
Mathematics Subject Classification: Primary: 76W05; Secondary: 35Q80, 70S15, 35Q60.
Citation: Jishan Fan, Fucai Li, Gen Nakamura. Low Mach number limit of the full compressible Hall-MHD system. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1731-1740. doi: 10.3934/cpaa.2017084
References:
[1]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[2]

W. Cui, Y. Ou and D. Ren, Incompressible limit of full compressible magnetohydrodynamic equations with well-prepared data in 3-D bounded domains, J. Math. Anal. Appl., 427 (2015), 263-288.  doi: 10.1016/j.jmaa.2015.02.049.

[3]

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.

[4]

D. Chae, P. Degond and J.-L. 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]

M. Dai, Regularity criterion for the 3D Hall-magneto-hydrodynamics, J. Differential Equations, 261 (2016), 573-591.  doi: 10.1016/j.jde.2016.03.019.

[7]

C. Dou, S. Jiang and Y. Ou, Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Differential Equations, 258 (2015), 379-398.  doi: 10.1016/j.jde.2014.09.017.

[8]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.

[9]

J. Fan, A. Alsaedi, T. Hayat, G. Nakamura and Y. Zhou, On strong solutions to the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 22 (2015), 423-434.  doi: 10.1016/j.nonrwa.2014.10.003.

[10]

J. Fan, B. Ahmad, T. Hayat and Y. Zhou, On well-posedness and blow-up for the full compressible Hall-MHD system, Nonlinear Anal. Real World Appl., 31 (2016), 569-579.  doi: 10.1016/j.nonrwa.2016.03.003.

[11]

J. Fan, Y. Gukuoto, G. 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.

[12]

J. Fan, F. 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.

[13]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001.

[14]

J. Fan and W. Yu, Global variational solutionis to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660.  doi: 10.1016/j.na.2007.10.005.

[15]

M. Fei and Z. Xiang, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics with horizontal dissipation J. Math. Phys. , 56 (2015), 051504, 13 pp. doi: 10.1063/1.4921653.

[16]

X. Hu and D. Wang, Global solutions to the three-demensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2.

[17]

X.-P. Hu and D.-H. Wang, Global existence and large-time behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[18]

S. Jiang, Q. Ju and F. Li, Low Mach number limit of the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365.  doi: 10.1088/0951-7715/25/5/1351.

[19]

S. Jiang, Q. Ju, F. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[20]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[21]

Y. Mu, Zero Mach number limit of the compressible Hall-magnetohydrodynamic equations Z. Angew. Math. Phys. , 67 (2016), Art. 1, 13 pp. doi: 10.1007/s00033-015-0604-0.

[22]

J. M. Polygiannakis and X. Moussas, A review of magneto-vorticity induction in Hall-MHD plasmas, Plasma Phys. Control. Fusion, 43 (2001), 195-221. 

[23]

D. Shaikh and P. K. Shukla, 3D simulations of fluctuation spectra in the Hall-MHD plasma, Phys. Rev. Lett. , 102 (2009), 045004, 4pp.

[24]

D. Shaikh and G. P. Zank, Spectral features of solar wind turbulent plasma, Monthly Notices of the Royal Astronomical Society, 400 (2009), 1881-1891. 

[25]

S. Servidioa, V. Carbonea, L. Primaveraa, P. Veltria and K. Stasiewicz, Compressible turbulence in Hall magnetohydrodynamics, Planet. Space Sci., 55 (2007), 2239-2243. 

[26]

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.

[27]

X. Yang, Low Mach number limit of the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 25 (2015), 118-126.  doi: 10.1016/j.nonrwa.2015.03.007.

[28]

W. M. Zajaczkowski, On nonstationary motion of a compressible barotropic viscous fluid with boundary slip condition, J. Appl. Anal., 4 (1998), 167-204.  doi: 10.1515/JAA.1998.167.

show all references

References:
[1]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[2]

W. Cui, Y. Ou and D. Ren, Incompressible limit of full compressible magnetohydrodynamic equations with well-prepared data in 3-D bounded domains, J. Math. Anal. Appl., 427 (2015), 263-288.  doi: 10.1016/j.jmaa.2015.02.049.

[3]

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.

[4]

D. Chae, P. Degond and J.-L. 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]

M. Dai, Regularity criterion for the 3D Hall-magneto-hydrodynamics, J. Differential Equations, 261 (2016), 573-591.  doi: 10.1016/j.jde.2016.03.019.

[7]

C. Dou, S. Jiang and Y. Ou, Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Differential Equations, 258 (2015), 379-398.  doi: 10.1016/j.jde.2014.09.017.

[8]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.

[9]

J. Fan, A. Alsaedi, T. Hayat, G. Nakamura and Y. Zhou, On strong solutions to the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 22 (2015), 423-434.  doi: 10.1016/j.nonrwa.2014.10.003.

[10]

J. Fan, B. Ahmad, T. Hayat and Y. Zhou, On well-posedness and blow-up for the full compressible Hall-MHD system, Nonlinear Anal. Real World Appl., 31 (2016), 569-579.  doi: 10.1016/j.nonrwa.2016.03.003.

[11]

J. Fan, Y. Gukuoto, G. 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.

[12]

J. Fan, F. 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.

[13]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001.

[14]

J. Fan and W. Yu, Global variational solutionis to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660.  doi: 10.1016/j.na.2007.10.005.

[15]

M. Fei and Z. Xiang, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics with horizontal dissipation J. Math. Phys. , 56 (2015), 051504, 13 pp. doi: 10.1063/1.4921653.

[16]

X. Hu and D. Wang, Global solutions to the three-demensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2.

[17]

X.-P. Hu and D.-H. Wang, Global existence and large-time behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[18]

S. Jiang, Q. Ju and F. Li, Low Mach number limit of the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365.  doi: 10.1088/0951-7715/25/5/1351.

[19]

S. Jiang, Q. Ju, F. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[20]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[21]

Y. Mu, Zero Mach number limit of the compressible Hall-magnetohydrodynamic equations Z. Angew. Math. Phys. , 67 (2016), Art. 1, 13 pp. doi: 10.1007/s00033-015-0604-0.

[22]

J. M. Polygiannakis and X. Moussas, A review of magneto-vorticity induction in Hall-MHD plasmas, Plasma Phys. Control. Fusion, 43 (2001), 195-221. 

[23]

D. Shaikh and P. K. Shukla, 3D simulations of fluctuation spectra in the Hall-MHD plasma, Phys. Rev. Lett. , 102 (2009), 045004, 4pp.

[24]

D. Shaikh and G. P. Zank, Spectral features of solar wind turbulent plasma, Monthly Notices of the Royal Astronomical Society, 400 (2009), 1881-1891. 

[25]

S. Servidioa, V. Carbonea, L. Primaveraa, P. Veltria and K. Stasiewicz, Compressible turbulence in Hall magnetohydrodynamics, Planet. Space Sci., 55 (2007), 2239-2243. 

[26]

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.

[27]

X. Yang, Low Mach number limit of the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 25 (2015), 118-126.  doi: 10.1016/j.nonrwa.2015.03.007.

[28]

W. M. Zajaczkowski, On nonstationary motion of a compressible barotropic viscous fluid with boundary slip condition, J. Appl. Anal., 4 (1998), 167-204.  doi: 10.1515/JAA.1998.167.

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