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March  2021, 20(3): 1297-1317. doi: 10.3934/cpaa.2021021

Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations

Huancheng Yao 1, , Haiyan Yin 2, and  Changjiang Zhu 1,,

1. 

School of Mathematics, South China University of Technology, Guangzhou 510641, China
2. 

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

* Corresponding author

Received  November 2020 Revised  December 2020 Published  February 2021

Fund Project: Huancheng Yao and Changjiang Zhu were supported by the National Natural Science Foundation of China #11771150, 11831003, 11926346 and Guangdong Basic and Applied Basic Research Foundation #2020B1515310015. Haiyan Yin was supported by the National Natural Science Foundation of China #12071163, 11601165 and the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University #ZQN-PY602

We study the large-time asymptotic behavior of solutions toward the rarefaction wave of the compressible non-isentropic Navier-Stokes equations coupling with Maxwell equations under some small perturbations of initial data and also under the assumption that the dielectric constant is bounded. For that, the dissipative structure of this hyperbolic-parabolic system is studied to include the effect of the electromagnetic field into the viscous fluid and turns out to be more complicated than that in the simpler compressible Navier-Stokes system. The proof of the main result is based on the elementary $ L^2 $ energy methods.

Keywords: Compressible non-isentropic Navier-Stokes-Maxwell equations, rarefaction wave, asymptotic behavior, electromagnetic field, dielectric constant.
Mathematics Subject Classification: Primary: 35Q30, 76N06, 76N30; Secondary: 35Q61.
Citation: Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021
References:
[1]

M. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4nd edition, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar

[2]

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

[3]

R. J. Duan, S. Q. Liu, H. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84.  doi: 10.1007/s11425-015-5059-4.  Google Scholar

[4]

J. S. Fan and Y. X. Hu, Uniform existence of the 1-d complete equations for an electromagnetic fluid, J. Math. Anal. Appl., 419 (2014), 1-9.  doi: 10.1016/j.jmaa.2014.04.052.  Google Scholar

[5]

J. S. Fan and Y. B. Ou, Uniform existence of the 1-D full equations for a thermo-radiative electromagnetic fluid, Nonlinear Anal., 106 (2014), 151-158.  doi: 10.1016/j.na.2014.04.018.  Google Scholar

[6]

F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.  doi: 10.1007/s00205-009-0267-0.  Google Scholar

[7]

F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.  doi: 10.1007/s00205-005-0380-7.  Google Scholar

[8]

F. M. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana Univ. Math. J., 65 (2016), 1833-1875.  doi: 10.1512/iumj.2016.65.5914.  Google Scholar

[9]

F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014.  Google Scholar

[10]

Y. T. Huang and H. X. Liu, Stability of rarefaction wave for a macroscopic model derived from the Vlasov-Maxwell-Boltzmann system, Acta Math. Sci. Ser. B, 38 (2018), 857-888.  doi: 10.1016/S0252-9602(18)30789-6.  Google Scholar

[11]

I. Imai, General Principles of Magneto-Fluid Dynamics. In: Magneto-Fulid Dynamics, Suppl. Prog. Theor. Phys., 24 (1962), 1-34.   Google Scholar

[12]

S. Jiang and F. C. 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

[13]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.  Google Scholar

[14]

S. Kawashima, Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869.  Google Scholar

[15]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127.   Google Scholar

[16]

S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. Math. Sci., 62 (1986), 249-252.   Google Scholar

[17]

S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic 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 electromagnetic fluid. II, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 181-184.   Google Scholar

[19]

T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108.  doi: 10.1090/memo/0328.  Google Scholar

[20]

T. P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Commun. Pure Appl. Math., 39 (1986), 565-594.  doi: 10.1002/cpa.3160390502.  Google Scholar

[21]

T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Commun. Math. Phys., 118 (1988), 451-465.   Google Scholar

[22]

F. Q. Luo, H. C. Yao and C. J. Zhu, Stability of rarefaction wave for isentropic compressible Navier-Stokes-Maxwell equations, Nonlinear Anal. Real World Appl., 59 (2021), 103234. doi: 10.1016/j.nonrwa.2020.103234.  Google Scholar

[23]

T. Luo, H. Y. Yin and C. J. Zhu, Stability of the composite wave for compressible Navier-Stokes/Allen-Cahn system, Math. Models Methods Appl. Sci., 30 (2020), 343-385.  doi: 10.1142/S0218202520500098.  Google Scholar

[24]

N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., 93 (2010), 559-571.  doi: 10.1016/j.matpur.2009.08.007.  Google Scholar

[25]

A. Matsumura, Waves in compressible fluids: viscous shock, rarefaction, and contact waves, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 2495–2548. doi: 10.1007/978-3-319-13344-7_60.  Google Scholar

[26]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088.  Google Scholar

[27] D. Mihalas and W. B. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, 1984.   Google Scholar
[28]

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

[29]

L. Z. Ruan, H. Y. Yin and C. J. Zhu, Stability of the superposition of rarefaction wave and contact discontinuity for the non-isentropic Navier-Stokes-Poisson system, Math. Methods Appl. Sci., 40 (2017), 2784-2810.  doi: 10.1002/mma.4198.  Google Scholar

[30]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[31]

X. Xu, Asymptotic behavior of solutions to an electromagnetic fluid model, Z. Angew. Math. Phys., 69 (2018), 1-19.  doi: 10.1007/s00033-018-0945-6.  Google Scholar

show all references

References:
[1]

M. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4nd edition, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar

[2]

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

[3]

R. J. Duan, S. Q. Liu, H. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84.  doi: 10.1007/s11425-015-5059-4.  Google Scholar

[4]

J. S. Fan and Y. X. Hu, Uniform existence of the 1-d complete equations for an electromagnetic fluid, J. Math. Anal. Appl., 419 (2014), 1-9.  doi: 10.1016/j.jmaa.2014.04.052.  Google Scholar

[5]

J. S. Fan and Y. B. Ou, Uniform existence of the 1-D full equations for a thermo-radiative electromagnetic fluid, Nonlinear Anal., 106 (2014), 151-158.  doi: 10.1016/j.na.2014.04.018.  Google Scholar

[6]

F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.  doi: 10.1007/s00205-009-0267-0.  Google Scholar

[7]

F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.  doi: 10.1007/s00205-005-0380-7.  Google Scholar

[8]

F. M. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana Univ. Math. J., 65 (2016), 1833-1875.  doi: 10.1512/iumj.2016.65.5914.  Google Scholar

[9]

F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014.  Google Scholar

[10]

Y. T. Huang and H. X. Liu, Stability of rarefaction wave for a macroscopic model derived from the Vlasov-Maxwell-Boltzmann system, Acta Math. Sci. Ser. B, 38 (2018), 857-888.  doi: 10.1016/S0252-9602(18)30789-6.  Google Scholar

[11]

I. Imai, General Principles of Magneto-Fluid Dynamics. In: Magneto-Fulid Dynamics, Suppl. Prog. Theor. Phys., 24 (1962), 1-34.   Google Scholar

[12]

S. Jiang and F. C. 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

[13]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.  Google Scholar

[14]

S. Kawashima, Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869.  Google Scholar

[15]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127.   Google Scholar

[16]

S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. Math. Sci., 62 (1986), 249-252.   Google Scholar

[17]

S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic 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 electromagnetic fluid. II, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 181-184.   Google Scholar

[19]

T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108.  doi: 10.1090/memo/0328.  Google Scholar

[20]

T. P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Commun. Pure Appl. Math., 39 (1986), 565-594.  doi: 10.1002/cpa.3160390502.  Google Scholar

[21]

T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Commun. Math. Phys., 118 (1988), 451-465.   Google Scholar

[22]

F. Q. Luo, H. C. Yao and C. J. Zhu, Stability of rarefaction wave for isentropic compressible Navier-Stokes-Maxwell equations, Nonlinear Anal. Real World Appl., 59 (2021), 103234. doi: 10.1016/j.nonrwa.2020.103234.  Google Scholar

[23]

T. Luo, H. Y. Yin and C. J. Zhu, Stability of the composite wave for compressible Navier-Stokes/Allen-Cahn system, Math. Models Methods Appl. Sci., 30 (2020), 343-385.  doi: 10.1142/S0218202520500098.  Google Scholar

[24]

N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., 93 (2010), 559-571.  doi: 10.1016/j.matpur.2009.08.007.  Google Scholar

[25]

A. Matsumura, Waves in compressible fluids: viscous shock, rarefaction, and contact waves, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 2495–2548. doi: 10.1007/978-3-319-13344-7_60.  Google Scholar

[26]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088.  Google Scholar

[27] D. Mihalas and W. B. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, 1984.   Google Scholar
[28]

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

[29]

L. Z. Ruan, H. Y. Yin and C. J. Zhu, Stability of the superposition of rarefaction wave and contact discontinuity for the non-isentropic Navier-Stokes-Poisson system, Math. Methods Appl. Sci., 40 (2017), 2784-2810.  doi: 10.1002/mma.4198.  Google Scholar

[30]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[31]

X. Xu, Asymptotic behavior of solutions to an electromagnetic fluid model, Z. Angew. Math. Phys., 69 (2018), 1-19.  doi: 10.1007/s00033-018-0945-6.  Google Scholar

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