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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 & 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. 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]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. 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]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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