Skip to main content

Asymptotic Stability of Combination of Viscous Contact Wave with Rarefaction Waves for One-Dimensional Compressible Navier–Stokes System

Abstract

We are concerned with the large-time behavior of solutions of the Cauchy problem to the one-dimensional compressible Navier–Stokes system for ideal polytropic fluids, where the far field states are prescribed. When the corresponding Riemann problem for the compressible Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is proved that for the one-dimensional compressible Navier–Stokes system, the combination wave of a “viscous contact wave”, which corresponds to the contact discontinuity, with rarefaction waves is asymptotically stable, provided the strength of the combination wave is suitably small. This result is proved by using elementary energy methods.

This is a preview of subscription content, access via your institution.

References

  1. 1.

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

    MATH  Article  Google Scholar 

  2. 2.

    Hsiao L., Liu T.: Nonlinear diffusive phenomena of nonlinear hyperbolic systems. Chinese Ann. Math. Ser. B 14, 465–480 (1993)

    MATH  MathSciNet  Google Scholar 

  3. 3.

    Huang F., Matsumur A., Shi X.: On the stability of contact discontinuity for compressible Navier–Stokes equations with free boundary. Osaka J. Math. 41, 193–210 (2004)

    MATH  MathSciNet  Google Scholar 

  4. 4.

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

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

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

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    Huang F., Zhao H.: On the global stability of contact discontinuity for compressible Navier–Stokes equations. Rend. Sem. Mat. Univ. Padova 109, 283–305 (2003)

    MATH  MathSciNet  Google Scholar 

  7. 7.

    Kawashima S., Matsumura A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Comm. Math. Phys. 101, 97–127 (1985)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  8. 8.

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

    MATH  Article  MathSciNet  Google Scholar 

  9. 9.

    Liu T.: Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Comm. Pure Appl. Math. 30, 767–796 (1977)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  10. 10.

    Liu T.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc. 56, 1–108 (1985)

    ADS  Google Scholar 

  11. 11.

    Liu T.: Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50, 1113–1182 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

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

    MATH  Article  MathSciNet  ADS  Google Scholar 

  13. 13.

    Liu T., Xin Z.: Pointwise decay to contact discontinuities for systems of viscous conservation laws. Asian J. Math. 1, 34–84 (1997)

    MATH  MathSciNet  Google Scholar 

  14. 14.

    Matsumura A., Nishihara K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Jpn J. Appl. Math. 2, 17–25 (1985)

    MATH  Article  MathSciNet  Google Scholar 

  15. 15.

    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)

    MATH  Article  MathSciNet  Google Scholar 

  16. 16.

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

    MATH  Article  MathSciNet  ADS  Google Scholar 

  17. 17.

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

    MATH  Article  MathSciNet  Google Scholar 

  18. 18.

    Smoller J.: Shock Waves and Reaction–Diffusion Equations, 2nd edn. Springer, New York (1994)

    MATH  Google Scholar 

  19. 19.

    Szepessy A., Xin Z.: Nonlinear stability of viscous shock waves. Arch. Ration. Mech. Anal. 122, 53–103 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  20. 20.

    Szepessy A., Zumbrun K.: Stability of rarefaction waves in viscous media. Arch. Ration. Mech. Anal. 133, 249–298 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  21. 21.

    van Duyn C.J., Peletier L.A.: A class of similarity solutions of the nonlinear diffusion equation. Nonlinear Anal. 1, 223–233 (1976/1977)

    Google Scholar 

  22. 22.

    Xin Z. et al.: On nonlinear stability of contact discontinuities. In: Glimm, J. (eds) Hyperbolic Problems: Theory, Numerics, Applications., pp. 249–257. World Scientific, River Edge (1996)

    Google Scholar 

  23. 23.

    Zeng H.: Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws. J. Differ. Equ. 246, 2081–2102 (2009)

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Feimin Huang.

Additional information

Communicated by T.-P. Liu

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Huang, F., Li, J. & Matsumura, A. Asymptotic Stability of Combination of Viscous Contact Wave with Rarefaction Waves for One-Dimensional Compressible Navier–Stokes System. Arch Rational Mech Anal 197, 89–116 (2010). https://doi.org/10.1007/s00205-009-0267-0

Download citation

Keywords

  • Cauchy Problem
  • Asymptotic Stability
  • Travel Wave Solution
  • Rarefaction Wave
  • Sobolev Inequality