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