Skip to main content
SpringerLink
Log in

Convergence to Steady-States of Compressible Navier–Stokes–Maxwell Equations

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

In this paper, we consider the compressible Navier–Stokes–Maxwell equations with a non-constant background density in \(\mathbb {R}^3\). We first show the existence and uniqueness of the non-trivial equilibrium (steady-state) of the system when the background density is a small variation of certain constant state, then we prove the asymptotic stability of the steady-state once the initial perturbation around the steady-state is small. Furthermore, by establishing the time-decay estimates for the corresponding linearized homogeneous equations, we artfully derive the time-algebraic convergence rates. The proof is based on the time-weighted energy method but with some new developments on the weight settings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Chen, G.Q., Jerome, J.W., Wang, D.H.: Compressible Euler-Maxwell equations. Transp. Statist. Phyd. 29, 311–331 (2000)

    Article  MathSciNet  Google Scholar 

  • Chen, Y., Li, F.C., Zhang, Z.P.: Large time behavior of the isentropic compressible Navier-Stokes-Maxwell system. Z. Angew. Math. Phys 67, 91 (2016)

    Article  MathSciNet  Google Scholar 

  • Deng, Y., Ionescu, A.D., Pausade, B.: The Euler-Maxwell system for electrons: global solutions in 2D. Arch. Rational Mech. Anal. 225, 771–871 (2017)

    Article  MathSciNet  Google Scholar 

  • Duan, R.J.: Global smooth flows for the compressible Euler-Maxwell system : the relaxation case. J. Hyper. Differ. Equ. 8, 375–413 (2011)

    Article  MathSciNet  Google Scholar 

  • Duan, R.J.: Green’s function and large time behavior of the Navier–Stokes-Maxwell system. Anal. Appl. 10, 133–197 (2012)

    Article  MathSciNet  Google Scholar 

  • Duan, R.J., Yang, T.: Stability of the one-species Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal. 41(10), 2353–2387 (2009)

    MathSciNet  MATH  Google Scholar 

  • Duan, R.J., Strain, R.M.: Optimal time decay of the Vlasov-Poisson-Boltzmann system in \(\mathbb{R}^3\). Arch. Rational Mech. Anal. 199, 291–328 (2011)

    Article  MathSciNet  Google Scholar 

  • Duan, R.J., Ukai, S., Yang, T., Zhao, H.J.: Optimal convergence rates for the compressible Navier–Stokes equations with potential forces. Math. Models Methods Appl. Sci. 17, 737–758 (2007)

    Article  MathSciNet  Google Scholar 

  • Duan, R.J., Liu, H.X., Ukai, S., Yang, T.: Optimal \(L^p-L^q\) convergence rates for the compressible Navier–Stokes equations with potential force. J. Differ. Equ. 238, 220–233 (2007)

    Article  Google Scholar 

  • Dumas, E., Lu, Y., Texier, B.: Space-time resonances and high-frequency Raman instabilities in the two-fluid Euler-Maxwell system, preprint (2021)

  • Fan, J., Li, F., Nakamura, G.: 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, 443–453 (2016)

    MATH  Google Scholar 

  • Feng, Y.H., Li, X., Mei, M., Wang, S.: Asymptotic decay of bipolar isentropic/non-isentropic compressible Navier–Stokes-Maxwell systems. J. Differ. Equ. 301, 471–542 (2021)

  • Feng, Y.H., Li, X., Wang, S.: Stability of non-constant equilibrium solutions for the full compressible Navier–Stokes-Maxwell system. J. Math. Fluid Mech. 23(2), 17 (2021)

  • Feng, Y.H., Wang, S., Kawashima, S.: Global existence and asymptotic decay of solutions to the non-isentropic Euler-Maxwell system. Math. Models Methods Appl. Sci. 24, 2851–2884 (2014)

  • Germain, P., Ibrahim, S., Masmoudi, N.: Wellposedness of the Navier-Stokes-Maxwell system. In: Proceedings of the Royal Society of Edinburgh, (2014)

  • Guo, Y., Ionescu, A.D., Pausader, B.: Global solutions of the Euler-Maxwell two fluid system in 3D. Ann. Math. 183(2), 377–498 (2016)

  • Germain, P., Masmoudi, N.: Global existence for the Euler-Maxwell system. Annales Scientifiques De L école Normale Supérieure 47(31), (2014)

  • Hajjej, M.L., Peng, Y.J.: Initial layers and zero-relaxation limits of Euler-Maxwell equations. J. Differ. Equ. 252, 1441–1465 (2012)

  • Ibrahim, S., Keraani, S.: Global small solutions of the Navier–Stokes-Maxwell equations. SIAM J. Math. Anal. 43, 2275–2295 (2011)

    Article  MathSciNet  Google Scholar 

  • Ibrahim, S., Yoneda, T.: Local solvablity and loss of smoothness of the Navier–Stokes-Maxwell equations with large initial data. J. Math. Anal. Appl. 2(396), 555–561 (2012)

    Article  Google Scholar 

  • Jerome, J.W.: The Cauchy problem for compressible hydrodynamic-Maxwell systems: a local theory for smooth solutions. Differ. Int. Equ. 16, 1345–1368 (2003)

    MathSciNet  MATH  Google Scholar 

  • Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rational Mech. Anal. 58, 181–205 (1975)

  • Krall, N.A., Trivelpiece, A.W.: Principles of Plasma Physics. San Francisco Press, Ny (1986)

    Google Scholar 

  • Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34, 481–524 (1981)

    Article  MathSciNet  Google Scholar 

  • Liu, Q.Q., Zhu, C.J.: Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations. Indiana. Univ. Math. J. 62, 1203–1235 (2013)

    Article  MathSciNet  Google Scholar 

  • Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer-Verlag, New York (1984)

    Book  Google Scholar 

  • Nicholson, D.R.: Introduction to Plasma Theory. Wiley, New York (1983)

    Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Matsumura, A., Nishida, T.: The initial value problem for the equation of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)

  • Peng, Y.J.: Stability of non-constant equilibrium solutions for Euler-Maxwell equations. J. Math. Pures Appl. 103, 39–67 (2015)

  • Peng, Y.J., Wang, S.: Convergence of compressible Euler-Maxwell equations to incompressible Euler equations. Comm. Partial Differ. Equ. 33, 349–376 (2008)

    Article  MathSciNet  Google Scholar 

  • Peng, Y.J., Wang, S., Gu, G.L.: Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations. SIAM J. Math. Anal. 43, 944–970 (2011)

    Article  MathSciNet  Google Scholar 

  • Ueda, Y., Wang, S., Kawashima, S.: Dissipative structure of the regularity type and time asymptotic decay of solutions for the Euler-Maxwell system. SIAM J. Math. Anal. 44, 2002–2017 (2012)

    Article  MathSciNet  Google Scholar 

  • Wang, W.K., Xu, X.: Large time behavior of solution for the full compressible Navier–Stokes-Maxwell system. Commun. Pure Appl. Anal. 14, 2283–2313 (2015)

  • Wasiolek, V.: Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Comm. Pure Appl. Anal. 15, 2007–2021 (2016)

    Article  MathSciNet  Google Scholar 

  • Xu, J.: Global classical solutions to the compressible Euler-Maxwell equations. SIAM J. Math. Anal. 43, 2688–2718 (2011)

    Article  MathSciNet  Google Scholar 

  • Yang, J.W., Wang, S.: The diffusive relaxation limit of non-isentropic Euler-Maxwell equations for plasmas. J. Math. Anal. Appl. 380, 343–353 (2011)

    Article  MathSciNet  Google Scholar 

  • Yang, Y.F., Hu, H.F.: Uniform global convergence of non-isentropic Euler-Maxwell system with dissipation. Nonlinear Anal. RealWorld 47, 332–347 (2019)

    Article  MathSciNet  Google Scholar 

  • Yue, G.C., Zhong, C.K.: On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Dis. Continu. Dynam. Syst 36(10), 5817–5835 (2016)

    Article  MathSciNet  Google Scholar 

  • Zhao, L.: The rigorous derivation of unipolar Euler-Maxwell system for electrons from bipolar Euler-Maxwell system by infinity-ion-mass limit. Math. Meth. Appl. Sci. 44(5), 3418–3440 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank two anonymous referees for their helpful comments and suggestions, which led to an important improvement of our original manuscript. This work was done when Y. Feng visited McGill University supported by China Scholarship Council (CSC) for the senior visiting scholar program (202006545001). He would like to express his sincere thanks for the hospitality of McGill University and CSC. The research of the authors was supported by the BNSF (1132006), NSFC (11831003, 11771031, 12171111, 12101060, 12171460), the project of the Beijing Education Committee (KZ202110005011), the general project of scientific research project of the Beijing Education Committee(KM202111232008), the research fund of Beijing Information Science and Technology University (2025029), and partially supported by NSERC grant RGPIN 354724-2016 and FRQNT grant 2019-CO-256440.

Author information

Authors and Affiliations

  1. College of Mathematics, Faculty of Science, Beijing University of Technology, Beijing, 100022, China

    Yue-Hong Feng, Shu Wang & Yang-Chen Cao

  2. College of Sciences, Beijing Information Science and Technology University, Beijing, 100192, China

    Xin Li

  3. Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada

    Ming Mei

  4. Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada

    Yue-Hong Feng & Ming Mei

Authors
  1. Yue-Hong Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ming Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yang-Chen Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yue-Hong Feng.

Additional information

Communicated by Lydia Bieri.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, YH., Li, X., Mei, M. et al. Convergence to Steady-States of Compressible Navier–Stokes–Maxwell Equations. J Nonlinear Sci 32, 2 (2022). https://doi.org/10.1007/s00332-021-09763-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-021-09763-9

Keywords

Mathematics Subject Classification

Search

Navigation