Skip to main content

Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compressible Navier-Stokes Equation

Abstract

In this paper we investigate the asymptotic stability of a composite wave consisting of two viscous shock waves for the full compressible Navier-Stokes equation. By introducing a new linear diffusion wave special to this case, we successfully prove that if the strengths of the viscous shock waves are suitably small with same order and also the initial perturbations which are not necessarily of zero integral are suitably small, the unique global solution in time to the full compressible Navier-Stokes equation exists and asymptotically tends toward the corresponding composite wave whose shifts (in space) of two viscous shock waves are uniquely determined by the initial perturbations. We then apply the idea to study a half space problem for the full compressible Navier-Stokes equation and obtain a similar result.

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

References

  1. 1

    Courant R., Friedrichs K.O.: Supersonic Flows and Shock Waves. Wiley-Interscience, New York (1948)

    Google Scholar 

  2. 2

    Goodman J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95(4), 325–344 (1986)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  3. 3

    Huang F.M., Matsumura A., Shi X.: On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary. Osaka J. Math. 41(1), 193–210 (2004)

    MATH  MathSciNet  Google Scholar 

  4. 4

    Huang F.M., Matsumura A.: Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations. Arch. Rat. Mech. Anal. 179, 55–77 (2005)

    Article  MathSciNet  Google Scholar 

  5. 5

    Huang F.M., Xin Z.P., Yang T.: Contact discontinuity with general perturbation for gas motion. Adv. Math. 219(4), 1246–1297 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6

    Huang F.M., Zhao H.J.: 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. Commun. Math. Phys. 101(1), 97–127 (1985)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  8. 8

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

  9. 9

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

    MATH  Article  ADS  MathSciNet  Google Scholar 

  10. 10

    Liu T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 56(329), 1–108 (1985)

    Google Scholar 

  11. 11

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

    MATH  Article  MathSciNet  Google Scholar 

  12. 12

    Liu T.-P.: Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50(11), 1113–1182 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  13. 13

    Liu T.-P., Xin Z.P.: Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Commun. Math. Phys. 118(3), 451–465 (1988)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  14. 14

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

    MATH  MathSciNet  Google Scholar 

  15. 15

    Matsumura A., Mei M.: Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Rat. Mech. Anal. 146, 1–22 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  16. 16

    Matsumura A., Nishihara K.: On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25 (1985)

    MATH  MathSciNet  Article  Google Scholar 

  17. 17

    Matsumura A., Nishihara K.: Asymptotics toward the rarefaction waves of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 3(1), 3–13 (1985)

    Google Scholar 

  18. 18

    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)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  19. 19

    Matsumura, A., Nishihara, K.: Global Solutions for Nonlinear Differential Equations–Mathematical Analysis on Compressible Viscous Fluids (In Japanese). Nippon Hyoronsha, 2004

  20. 20

    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)

    MATH  Article  MathSciNet  Google Scholar 

  21. 21

    Smoller J.: Shock Waves and Reaction-Diffusion Equations, Second Edition. Springer-Verlag, New York (1994)

    Google Scholar 

  22. 22

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

    MATH  Article  MathSciNet  Google Scholar 

  23. 23

    Xin, Z.P.: On nonlinear stability of contact discontinuities. In: Hyperbolic Problems: Theory, Numerics, Applications (Stony Brook, NY, 1994), River Edge, NJ: World Sci. Publishing, 1996, pp. 249–257

  24. 24

    Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In: Handbook of Mathematical Fluid Dynamics, Vol. III, Amsterdam: North-Holland, 2004, pp. 311–533

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Feimin Huang.

Additional information

Research is supported in part by NSFC Grant No. 10471138, NSFC-NSAF Grant No. 10676037 and 973 project of China, Grant No. 2006CB805902, in part by Japan Society for the Promotion of Science, the Invitation Fellowship for Research in Japan (Short-Term).

Research is supported in part by Grant-in-Aid for Scientific Research (B) 19340037, Japan.

Communicated by P. Constantin

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Huang, F., Matsumura, A. Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compressible Navier-Stokes Equation. Commun. Math. Phys. 289, 841–861 (2009). https://doi.org/10.1007/s00220-009-0843-z

Download citation

Keywords

  • Shock Wave
  • Asymptotic Stability
  • Travel Wave Solution
  • Rarefaction Wave
  • Riemann Problem