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February  2018, 11(1): 97-106. doi: 10.3934/krm.2018005

Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum

1. 

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

2. 

Institute of Applied Physics and Computational Mathematics, FengHao East Road, Haidian District, Beijing 100094, China

* Corresponding author: Yueling Jia

Received  December 2016 Revised  March 2017 Published  August 2017

Fund Project: This work is supported by NSFC grant No.11171154, 11271051.

In this paper, we prove the local well-posedness of strong solutions for a compressible Navier-Stokes-Maxwell system, provided the initial data satisfy a natural compatibility condition. We do not assume the positivity of initial density, it may vanish in an open subset (vacuum) of $Ω$.

Citation: Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic and Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005
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]

Y. Cho and H. Kim, Existence results for viscous polytropic fluid with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.

[4]

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

[5]

R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.  doi: 10.1142/S0219530512500078.

[6]

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

[7]

J. FanF. Li and G. Nakamura, Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydarodynamic equations in a bounded domain, Kinet. Relat. Models, 9 (2016), 443-453.  doi: 10.3934/krm.2016002.

[8]

J. FanF. Li and G. Nakamura, Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydarodynamic equations in a bounded domain Ⅱ: global existence case, J. Math. Fluid Mech., 9 (2016), 443-453.  doi: 10.3934/krm.2016002.

[9]

Y. H. FengS. Wang and X. Li, Asymptotic behavior of global smooth solutions for bipolar compressible Navier-Stokes-Maxwell system from plasmas, Acta Mathematica Scientia, 35 (2015), 955-969.  doi: 10.1016/S0252-9602(15)30030-8.

[10]

G. Y. HongX. F. HouH. Y. Peng and C. J. Zhu, Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum, Sci. China Math., 57 (2014), 2463-2484.  doi: 10.1007/s11425-014-4896-x.

[11]

X. Hou and L. Zhu, Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum, Commun. Pure Appl. Anal., 15 (2016), 161-183.  doi: 10.3934/cpaa.2016.15.161.

[12]

X. F. HouL. Yao and C. J. Zhu, Existence and uniqueness of global strong solutions to the Navier-Stokes-Maxwell system with large initial data and vacuum, Scientia Sinica Mathematica, 46 (2016), 945-966. 

[13]

I. Imai, General Principles of Magneto-Fluid Dynamics in "Magneto-Fluid Dynamics, " Suppl. Prog. Theor. Phys. 24(ed. H. Yukawa), Chap. I, RIFP Kyoto Univ. , 1962.

[14]

S. Jiang and F. C. Li, Converagese of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asymptotic Analysis, 95 (2015), 161-185.  doi: 10.3233/ASY-151321.

[15]

S. Jiang and F. C. 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.

[16]

E. Kang and J. Lee, Notes on the global well-posedness for the Maxwell-Navier-Stokes system, Abstract and Applied Analysis, 2013 (2013), Art. ID 402793, 6 pp.

[17]

S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an eletromagnetic fluid, Tsukuba J. Math., 10 (1986), 131-149.  doi: 10.21099/tkbjm/1496160397.

[18]

S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an eletromagnetic fluid Ⅱ, Proc. Japan Acad., 62 (1986), 181-184.  doi: 10.3792/pjaa.62.181.

[19]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics San Francisco Press, 1986.

[20]

F. C. 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.

[21]

E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, AMS, 2001. doi: 10.1090/gsm/014.

[22]

P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2, Compressible Models, Oxford University Press, New York, 1998.

[23]

Q. Q. Liu and Y. F. Su, Large time behavior for the non-isentropic Navier-Stokes-Maxwell system, Mathematical Methods in the Applied Sciences, 40 (2017), 663-679.  doi: 10.1002/mma.3999.

[24]

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.

[25]

S. -I. Pai, Magnetogasdynamics and Plasma Dynamics, Springer-Verlag, Vienna, 1962.

[26]

W. K. Wang and X. Xu, Large time behavior of solution for the full compressible navier-stokes-maxwell system, Commun. Pure Appl. Anal., 14 (2015), 2283-2313.  doi: 10.3934/cpaa.2015.14.2283.

[27]

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.

[28]

W. M. Zajaczkowski, On nonstationary motioni 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]

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]

Y. Cho and H. Kim, Existence results for viscous polytropic fluid with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.

[4]

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

[5]

R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.  doi: 10.1142/S0219530512500078.

[6]

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

[7]

J. FanF. Li and G. Nakamura, Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydarodynamic equations in a bounded domain, Kinet. Relat. Models, 9 (2016), 443-453.  doi: 10.3934/krm.2016002.

[8]

J. FanF. Li and G. Nakamura, Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydarodynamic equations in a bounded domain Ⅱ: global existence case, J. Math. Fluid Mech., 9 (2016), 443-453.  doi: 10.3934/krm.2016002.

[9]

Y. H. FengS. Wang and X. Li, Asymptotic behavior of global smooth solutions for bipolar compressible Navier-Stokes-Maxwell system from plasmas, Acta Mathematica Scientia, 35 (2015), 955-969.  doi: 10.1016/S0252-9602(15)30030-8.

[10]

G. Y. HongX. F. HouH. Y. Peng and C. J. Zhu, Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum, Sci. China Math., 57 (2014), 2463-2484.  doi: 10.1007/s11425-014-4896-x.

[11]

X. Hou and L. Zhu, Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum, Commun. Pure Appl. Anal., 15 (2016), 161-183.  doi: 10.3934/cpaa.2016.15.161.

[12]

X. F. HouL. Yao and C. J. Zhu, Existence and uniqueness of global strong solutions to the Navier-Stokes-Maxwell system with large initial data and vacuum, Scientia Sinica Mathematica, 46 (2016), 945-966. 

[13]

I. Imai, General Principles of Magneto-Fluid Dynamics in "Magneto-Fluid Dynamics, " Suppl. Prog. Theor. Phys. 24(ed. H. Yukawa), Chap. I, RIFP Kyoto Univ. , 1962.

[14]

S. Jiang and F. C. Li, Converagese of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asymptotic Analysis, 95 (2015), 161-185.  doi: 10.3233/ASY-151321.

[15]

S. Jiang and F. C. 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.

[16]

E. Kang and J. Lee, Notes on the global well-posedness for the Maxwell-Navier-Stokes system, Abstract and Applied Analysis, 2013 (2013), Art. ID 402793, 6 pp.

[17]

S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an eletromagnetic fluid, Tsukuba J. Math., 10 (1986), 131-149.  doi: 10.21099/tkbjm/1496160397.

[18]

S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an eletromagnetic fluid Ⅱ, Proc. Japan Acad., 62 (1986), 181-184.  doi: 10.3792/pjaa.62.181.

[19]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics San Francisco Press, 1986.

[20]

F. C. 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.

[21]

E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, AMS, 2001. doi: 10.1090/gsm/014.

[22]

P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2, Compressible Models, Oxford University Press, New York, 1998.

[23]

Q. Q. Liu and Y. F. Su, Large time behavior for the non-isentropic Navier-Stokes-Maxwell system, Mathematical Methods in the Applied Sciences, 40 (2017), 663-679.  doi: 10.1002/mma.3999.

[24]

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.

[25]

S. -I. Pai, Magnetogasdynamics and Plasma Dynamics, Springer-Verlag, Vienna, 1962.

[26]

W. K. Wang and X. Xu, Large time behavior of solution for the full compressible navier-stokes-maxwell system, Commun. Pure Appl. Anal., 14 (2015), 2283-2313.  doi: 10.3934/cpaa.2015.14.2283.

[27]

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.

[28]

W. M. Zajaczkowski, On nonstationary motioni 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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