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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[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.   Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[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.   Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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