Skip to main content
SpringerLink
Log in

Nonlinear stability of rarefaction waves for one-dimensional compressible Navier–Stokes equations for a reacting mixture

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we study the long-time behavior toward rarefaction waves for the Cauchy problem to a one-dimensional Navier–Stokes equations for a reacting mixture. It is shown that under the condition adiabatic exponent \(\gamma \) is close to 1, the global stability is established. In this paper, the initial perturbation can be large.

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

  1. Chen, G.Q.: Global solutions to the compressible Navier–Stokes equations for a reacting mixture. SIAM J. Math. Anal. 23, 609–634 (1992)

    Article  MathSciNet  Google Scholar 

  2. Chen, G.Q., Hoff, D., Trivisa, M.: Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data. Arch. Ration. Mech. Anal. 166, 321–358 (2003)

    Article  MathSciNet  Google Scholar 

  3. Chen, G.Q., Wagner, D.H.: Global entropy solutions to exothermically reacting, compressible Euler equations. J. Differ. Equ. 191, 277–322 (2003)

    Article  MathSciNet  Google Scholar 

  4. Cho, Y., Kim, H.: Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228, 377–411 (2006)

    Article  MathSciNet  Google Scholar 

  5. Donatelli, D., Trivisa, K.: On the motion of a viscous compressible radiative-reacting gas. Commun. Math. Phys. 265, 463–491 (2006)

    Article  MathSciNet  Google Scholar 

  6. Documet, B., Zlotnik, A.: Lyapunov functional method for 1D radiative and reactive viscous gas dynamics. Arch. Ration. Mech. Anal. 177, 185–229 (2005)

    Article  MathSciNet  Google Scholar 

  7. Ding, S.J., Wen, H.Y., Zhu, C.J.: Global classical large solutions to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum. J. Differ. Equ. 251, 1696–1725 (2011)

    Article  MathSciNet  Google Scholar 

  8. Duan, R.J., Liu, S.Q., Yin, H.Y., Zhu, C.J.: Stability of the rarefaction wave for a two-fluid plasma model with diffusion. Sci. China Ser. A 59, 67–84 (2016)

    Article  MathSciNet  Google Scholar 

  9. Feireisl, E.: On the motion of a viscous, compressible and heat-conducting fluid. Indiana Univ. Math. J. 53, 1705–1738 (2004)

    Article  MathSciNet  Google Scholar 

  10. Gardnea, R.A.: On the detonation of a combustible gas. Trans. Am. Math. Soc. 277, 431–468 (1983)

    Article  MathSciNet  Google Scholar 

  11. Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Ration. Mech. Anal. 95, 325–344 (1986)

    Article  MathSciNet  Google Scholar 

  12. Hoff, D.: Global existence for 1D, compressible, isentropic Navier–Stokes equations with large initial data. Trans. Am. Math. Soc. 303(1), 169–181 (1987)

    MATH  Google Scholar 

  13. Huang, F.M., Li, J., Matsumura, A.: Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier–Stokes system. Arch. Ration. Mech. Anal. 197, 89–116 (2010)

    Article  MathSciNet  Google Scholar 

  14. Huang, F.M., Matsumura, A., Xin, Z.P.: Stability of contact discontinuities for the 1-D compressible Navier–Stokes equations. Arch. Ration. Mech. Anal. 179, 55–77 (2006)

    Article  MathSciNet  Google Scholar 

  15. Huang, F.M., Xin, Z.P., Yang, T.: Contact discontinuity with general perturbations for gas motions. Adv. Math. 219, 1246–1297 (2008)

    Article  MathSciNet  Google Scholar 

  16. Huang, X.D., Li, J., Xin, Z.P.: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier–Stokes equations. Commun. Pure Appl. Math. 65, 549–585 (2012)

    Article  MathSciNet  Google Scholar 

  17. Jiang, S.: Large-time behavior of solutions to the equations of a viscous polytropic ideal gas. Ann. Mat. Pura Appl. 4, 253–275 (1998)

    Article  MathSciNet  Google Scholar 

  18. Jiang, S.: Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains. Commun. Math. Phys. 200, 181–193 (1999)

    Article  MathSciNet  Google Scholar 

  19. Jiang, S.: Remarks on the asymptotic begavior of solutions to the compressible Navier–Stokes equations in the half-line. Proc. R. Soc. Edinb. Sect. A 132, 627–638 (2002)

    Article  Google Scholar 

  20. Kanel’, Y.: On a model system of equations of one-dimensional gas motion. Differ. Uravn. 4, 374–380 (1968)

    MathSciNet  MATH  Google Scholar 

  21. Kazhikhov, A.V.: Cauchy problem for viscous gas equations gas equations. Siber. Math. J. 23, 44–49 (1982)

    Article  MathSciNet  Google Scholar 

  22. Kazhikhov, A.V., Shelukhin, V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, 273–282 (1977)

    Article  MathSciNet  Google Scholar 

  23. Li, J., Liang, Z.L.: Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier–Stokes system in unbounded domains with large data. Arch. Ration. Mech. Anal. 220, 1195–1208 (2016)

    Article  MathSciNet  Google Scholar 

  24. Li, S.R.: On one-dimensional compressible Navier–Stokes equations for a reacting mixture in unbounded domains. Z. Angew. Math. Phys. 68, 106 (2017)

    Article  MathSciNet  Google Scholar 

  25. Liao, Y.K., Zhao, H.J.: Global existence and large-time behavior of solutions to the Czuchy problem of one-dimensional viscous radiative and reactive gas. J. Differ. Equ. 265, 2076–2120 (2018)

    Article  Google Scholar 

  26. Liu, S.Q., Yin, H.Y., Zhu, C.J.: Stability of contact discontinuity for the Navier–Stokes–Poisson system with free boundary. Commun. Math. Sci. 14, 1859–1887 (2016)

    Article  MathSciNet  Google Scholar 

  27. Liu, T.P.: Shock waves for compressible Navier–Stokes equations are stable. Commun. Pure. Appl. Math. 39, 565–594 (1986)

    Article  MathSciNet  Google Scholar 

  28. Liu, T.P., Xin, Z.P.: Nonlinear stability of rarefaction waves for compressible Navier–Stokes equations. Commun. Math. Phys. 118, 451–465 (1988)

    Article  MathSciNet  Google Scholar 

  29. Luo, T., Yin, H.Y., Zhu, C.J.: Stability of the rarefaction wave for a coupled compressible Navier–Stokes/Allen–Cahn system. Math. Methods Appl. Sci. 41, 4724–4736 (2018)

    Article  MathSciNet  Google Scholar 

  30. Matsumura, A., Nishihara, K.: Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. Jpn. J. Appl. Math. 3, 1–13 (1986)

    Article  MathSciNet  Google Scholar 

  31. Matsumura, A., Nishihara, K.: Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Commun. Math. Phys. 144, 325–335 (1992)

    Article  MathSciNet  Google Scholar 

  32. Nishihara, K., Yang, T., Zhao, H.J.: Nonlinear stability of strong rarefaction waves for compressible Navier–Stokes equations. SIAM. J. Math. Anal. 35(6), 1561–1597 (2004)

    Article  MathSciNet  Google Scholar 

  33. Ruan, L.Z., Yin, H.Y., Zhu, C.J.: Stability of the superposition of rarefaction wave and contact discontinuity for the non-isentropic Navier–Stokes–Poisson equations with free boundary. Math. Methods Appl. Sci. 40, 2784–2810 (2017)

    Article  MathSciNet  Google Scholar 

  34. Tang, S.J., Zhang, L.: Nonlinear stability of viscous shock waves for one-dimensional nonisentropic compressible Navier–Stokes equations with a class of large initial perturebation. Acta. Math. Sci. 48B(3), 973–1000 (2018)

    Article  Google Scholar 

  35. Wanger, D.H.: The existence and behavior of viscous structure for plane detonation waves. SIAM J. Math. Anal. 20, 1035–1054 (1989)

    Article  MathSciNet  Google Scholar 

  36. Wang, D.H.: Global solutions for the mixture of real compressible reacting flows in combustion. Commun. Pure. Appl. Anal. 3, 775–790 (2004)

    Article  MathSciNet  Google Scholar 

  37. Williams, F.A.: Combustion Theory. Addison-Wesley, Reading (1965)

    Google Scholar 

  38. Yin, H.Y., Zhang, J.S., Zhu, C.J.: Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier–Stokes–Poisson system. Nonlinear Anal. Real World Appl. 14, 735–776 (2016)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Professor Changjiang Zhu for his sincere suggestions. The authors are supported by the National Natural Science Foundation of China \(\sharp \,11771150\) and \(\sharp \,11331005\), the Fundamental Research Funds for the Central Universities (No. D2172260) and Grant from SCUT (No. D6182820).

Author information

Authors and Affiliations

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

    Zheng Xu & Zefu Feng

Authors
  1. Zheng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zefu Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zefu Feng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, Z., Feng, Z. Nonlinear stability of rarefaction waves for one-dimensional compressible Navier–Stokes equations for a reacting mixture. Z. Angew. Math. Phys. 70, 155 (2019). https://doi.org/10.1007/s00033-019-1201-4

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-019-1201-4

Keywords

Mathematics Subject Classification

Search

Navigation