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September  2016, 9(3): 443-453. doi: 10.3934/krm.2016002

## Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain

 1 Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037 2 Department of Mathematics, Nanjing University, Nanjing 210093 3 Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

Received  April 2015 Revised  October 2015 Published  May 2016

In this paper we establish the uniform estimates of strong solutions with respect to the Mach number and the dielectric constant to the full compressible Navier-Stokes-Maxwell system in a bounded domain. Based on these uniform estimates, we obtain the convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations for well-prepared data.
Citation: Jishan Fan, Fucai Li, Gen Nakamura. Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain. Kinetic and Related Models, 2016, 9 (3) : 443-453. doi: 10.3934/krm.2016002
##### 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] J. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Funct. Anal., 15 (1974), 341-363. doi: 10.1016/0022-1236(74)90027-5. [3] 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. [4] 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. [5] J. Fan, F. Li and G. Nakamura, Uniform well-posedness and singular limits of the isentropic Navier-Stokes-Maxwell system in a bounded domain, Z. Angew. Math. Phys., 66 (2015), 1581-1593. doi: 10.1007/s00033-014-0484-8. [6] I. Imai, General principles of magneto-fluid dynamics, in Magneto-Fluid Dynamics, Suppl. Prog. Theor. Phys. No.24 (ed. H. Yukawa), RIFP Kyoto Univ., (1962), Chap. I, 1-34. [7] S. Jiang and F. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asymptot. Anal., 95 (2015), 161-185. doi: 10.3233/ASY-151321. [8] S. Jiang and F. Li, Zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system, Sci. China Math., 58 (2015), 61-76. doi: 10.1007/s11425-014-4923-y. [9] S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math., 10 (1986), 131-149. [10] S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid II, Proc. Japan Acad., Ser. A, 62 (1986), 181-184. doi: 10.3792/pjaa.62.181. [11] F. Li and Y. Mu, Low Mach number limit of the full compressible Navier-Stokes-Maxwell system, J. Math. Anal. Appl., 412 (2014), 334-344. doi: 10.1016/j.jmaa.2013.10.064. [12] G. Metivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90. doi: 10.1007/PL00004241. [13] Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055. doi: 10.1002/cpa.20187. [14] 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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##### 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] J. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Funct. Anal., 15 (1974), 341-363. doi: 10.1016/0022-1236(74)90027-5. [3] 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. [4] 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. [5] J. Fan, F. Li and G. Nakamura, Uniform well-posedness and singular limits of the isentropic Navier-Stokes-Maxwell system in a bounded domain, Z. Angew. Math. Phys., 66 (2015), 1581-1593. doi: 10.1007/s00033-014-0484-8. [6] I. Imai, General principles of magneto-fluid dynamics, in Magneto-Fluid Dynamics, Suppl. Prog. Theor. Phys. No.24 (ed. H. Yukawa), RIFP Kyoto Univ., (1962), Chap. I, 1-34. [7] S. Jiang and F. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asymptot. Anal., 95 (2015), 161-185. doi: 10.3233/ASY-151321. [8] S. Jiang and F. Li, Zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system, Sci. China Math., 58 (2015), 61-76. doi: 10.1007/s11425-014-4923-y. [9] S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math., 10 (1986), 131-149. [10] S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid II, Proc. Japan Acad., Ser. A, 62 (1986), 181-184. doi: 10.3792/pjaa.62.181. [11] F. Li and Y. Mu, Low Mach number limit of the full compressible Navier-Stokes-Maxwell system, J. Math. Anal. Appl., 412 (2014), 334-344. doi: 10.1016/j.jmaa.2013.10.064. [12] G. Metivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90. doi: 10.1007/PL00004241. [13] Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055. doi: 10.1002/cpa.20187. [14] 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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