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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

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

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.
Keywords: Full compressible Navier-Stokes-Maxwell system, incompressible magnetohydrodynamic equations, bounded domain., zero Mach number limit, zero dielectric constant limit.
Mathematics Subject Classification: Primary: 76W05; Secondary: 35Q60, 35B2.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math., 10 (1986), 131-149.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math., 10 (1986), 131-149.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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