Abstract
We study the stability of smooth solutions near non-constant equilibrium states for a bipolar full compressible Navier–Stokes–Maxwell system in a three-dimensional torus \(\mathbb {T}= (\mathbb {R}/\mathbb {Z})^3\). This system is quasilinear hyperbolic-parabolic. In the first part, by using the maximum principle, we find a non-constant steady state solution with small amplitude for this system. In the second part, with the help of suitable choices of symmetrizers and classic energy estimates, we prove that global smooth solutions exist and converge to the non-constant steady states as the time goes to infinity. As a byproduct, we obtain the global stability for the bipolar full compressible Navier–Stokes–Poisson system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, F.: Introduction to Plasma Physics and Controlled Fusion, vol. 1. Plenum Press, New York (1984)
Duan, R.J.: Global smooth flows for the compressible Euler–Maxwell system: relaxation case. J. Hyperb. Differ. Equ. 8, 375–413 (2011a)
Duan, R.J.: Green’s function and large time behavior of the Navier–Stokes–Maxwell system. Anal. Appl. 10(2), 133–197 (2011b)
Duan, R.J., Liu, Q.Q., Zhu, C.J.: Darcy’s law and diffusion for a two-fluid Euler–Maxwell system with dissipation. Math. Models Methods Appl. Sci. 25(11), 2089–2151 (2015)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)
Feng, Y.H., Peng, Y.J., Wang, S.: Asymptotic behavior of global smooth solutions for full compressible Navier–Stokes–Maxwell equations. Nonlinear Anal. Real 10, 105–116 (2014)
Feng, Y.H., Peng, Y.J., Wang, S.: Stability of non-constant equilibrium solutions for two-fluid Euler–Maxwell systems. Nonlinear Anal. Real 26, 372–390 (2015a)
Feng, Y.H., Wang, S., Li, X.: Asymptotic behavior of global smooth solutions for bipolar compressible Navier–Stokes–Maxwell system from plasmas. Acta Math. Sci. Ser. B 35B(5), 955–969 (2015b)
Feng, Y.H., Wang, S., Li, X.: Stability of non-constant steady-state solutions for non-isentropic Euler–Maxwell system with a temperature damping term. Math. Methods Appl. Sci. (2015c) Online
Germain, P., Ibrahim, S., Masmoudi, N.: Wellposedness of the Navier–Stokes–Maxwell equations. arxiv:1207.6187v1, (2012)
Hao, C., Li, H.L.: Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions. J. Differ. Equ. 246, 4791–4812 (2009)
Hsiao, L., Li, H.L., Yang, T., Zou, C.: Compressible non-isentropic bipolar Navier–Stokes–Poisson system in \(\mathbb{R}^3\). Acta Math. Sci. Ser. B 31B(6), 2169–2914 (2011)
Ibrahim, S., Yoneda, T.: Local solvability and loss of smoothness of the Navier–Stokes–Maxwell equations with large initial data. J. Math. Anal. Appl. 396(2), 555–561 (2012)
Ibrahim, S., Keraani, S.: Global small solutions of the Navier–Stokes–Maxwell equations. SIAM J. Math. Anal. 43, 2275–2295 (2011)
Ju, Q.C., Li, F.C., Li, H.L.: The quasineutral limit of Navier–Stokes–Poisson system with heat conductivity and general initial data. J. Differ. Equ. 247, 203–224 (2009)
Jüngel, A.: Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, Basel (2001)
Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975)
Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)
Li, H.L., Matsumura, A., Zhang, G.: Optimal decay rate of the compressible Navier–Stokes–Poisson system in \(\mathbb{R}^3\). Arch. Ration. Mech. Anal. 196, 681–713 (2010)
Liu, Q.Q., Zhu, C.J.: Asymptotic stability of stationary solutions to the compressible Euler–Maxwell equations. Indiana Univ. Math. J. 62(4), 1203–1235 (2013)
Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)
Markowich, P., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer, Berlin (1990)
Matsumura, A., Nishida, T.: The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A 55, 337–342 (1979)
Nishida, T.: Nonlinear hyperbolic equations and related topics in fluids dynamics. Publications Mathématiques d’Orsay, Université Paris-Sud, Orsay, No. 78–02, (1978)
Peng, Y.J.: Stability of non-constant equilibrium solutions for Euler–Maxwell equations. J. Math. Pures Appl. 103, 39–67 (2015)
Rishbeth, H., Garriott, O.K.: Introduction to Ionospheric Physics. Academic Press, Cambridge (1969)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series. Princeton University Press, Princeton (1970)
Tan, Z., Wang, Y.J., Wang, Y.: Stability of steady states of the Navier–Stokes–Poisson equations with non-flat doping profile. SIAM J. Math. Anal. 47(1), 179–209 (2015)
Ueda, Y., Kawashima, S.: Asymptotic stability of stationary solutions for the non-isentropic Euler–Maxwell system. Math. Anal. Fluid Gas Dyn. 2014, 13–20 (1883)
Wang, S., Jiang, S.: The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ. 31, 571–591 (2006)
Wang, W., Wu, Z.: Pointwise estimates of solution for the non-isentropic Navier–Stokes–Poisson equations in multi-dimensions. J. Differ. Equ. 248, 1617–1636 (2010)
Zhang, Y., Tan, Z.: On the existence of solutions to the Navier–Stokes–Poisson equations of a two-dimensional compressible flow. Math. Methods Appl. Sci. 30(3), 305–329 (2007)
Zhang, G., Li, H.L., Zhu, C.: Optimal decay rate of the non-isentropic compressible Navier–Stokes–Poisson system in \(\mathbb{R}^3\). J. Differ. Equ. 250(2), 866–891 (2011)
Acknowledgements
The authors are grateful to the referee for the comments. The third author would like to express his sincere gratitude to Professor Yue-Jun Peng of Université Blaise Pascal for excellent directions in France. The authors are supported by the the BNSF (1164010, 1132006), NSFC (11771031, 11371042), NSF of Qinghai Province (2017-ZJ-908), NSF of Henan Province (162300410084), the Key Research Fund of Henan Province (16A110019), International Research Seed Fund Project of BJUT, the key fund of the Beijing education committee of China, the general project of scientific research project of the Beijing education committee of China, the collaborative innovation center on Beijing society-building and social governance, the China postdoctoral science foundation funded project, the Project supported by Beijing Postdoctoral Research Foundation, the government of Chaoyang district postdoctoral research foundation, the 2016 Beijing project of scientific activities for the excellent students studying abroad and the Beijing University of Technology foundation funded project.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Alex Kiselev.
Rights and permissions
About this article
Cite this article
Li, X., Wang, S. & Feng, YH. Stability of Non-constant Equilibrium Solutions for Bipolar Full Compressible Navier–Stokes–Maxwell Systems. J Nonlinear Sci 28, 2187–2215 (2018). https://doi.org/10.1007/s00332-017-9435-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-017-9435-9