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

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

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  March 2021 Early access  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.

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. DuanS. Q. LiuH. 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. HuangJ. 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. HuangA. 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. HuangZ. 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. KawashimaA. 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. LuoH. 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. RuanH. 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. DuanS. Q. LiuH. 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. HuangJ. 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. HuangA. 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. HuangZ. 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. KawashimaA. 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. LuoH. 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. RuanH. 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

[1]

Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic & Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010

[2]

Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. doi: 10.3934/dcds.2021005

[3]

Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615

[4]

Xueke Pu, Min Li. Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5149-5181. doi: 10.3934/dcdsb.2019055

[5]

Weike Wang, Xin Xu. Large time behavior of solution for the full compressible navier-stokes-maxwell system. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2283-2313. doi: 10.3934/cpaa.2015.14.2283

[6]

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

[7]

Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477

[8]

Xiaofeng Hou, Limei Zhu. Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum. Communications on Pure & Applied Analysis, 2016, 15 (1) : 161-183. doi: 10.3934/cpaa.2016.15.161

[9]

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

[10]

Lvqiao liu, Lan Zhang. Optimal decay to the non-isentropic compressible micropolar fluids. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4575-4598. doi: 10.3934/cpaa.2020207

[11]

Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009

[12]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263

[13]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052

[14]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

[15]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004

[16]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985

[17]

Hong Cai, Zhong Tan, Qiuju Xu. Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type. Kinetic & Related Models, 2015, 8 (1) : 29-51. doi: 10.3934/krm.2015.8.29

[18]

Jing Wang, Feng Xie. On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2767-2784. doi: 10.3934/dcdsb.2016072

[19]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056

[20]

Linglong Du, Haitao Wang. Pointwise wave behavior of the Navier-Stokes equations in half space. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1349-1363. doi: 10.3934/dcds.2018055

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (131)
  • HTML views (105)
  • Cited by (0)

Other articles
by authors

[Back to Top]