Skip to main content
SpringerLink
Log in

Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed byv t −u x = 0,u t +p(v) x = μ(u x/v) x onR 1+ with boundary. The initial data of (v,u) have constant states (v +,u+) at +∞ and the boundary condition atx = 0 is given only on the velocityu, say u−. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, whenu − <u +,v − is determined as (u +,u +) ∈R 2(v −,u −), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction wave (v r,ur)(x/t) connecting two constant states (v −,u −) and (v +,u +). Our assertion is that the solution of the original system tends to the restriction of (v r,ur)(x/t) toR 1+ as t → ∞ provided that both the initial perturbations and ¦(v + −v −,u +-u t-) are small. The result is given by an elementaryL 2 energy method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Y. Hattori and K. Nishihara, A note on the stability of rarefaction wave of Burgers equation. Japan J. Indust. Appl. Math.,8 (1991), 85–96.

    Article  MATH  MathSciNet  Google Scholar 

  2. A.M. Ilin and O.A. Oleinik, Asymptotic behavior of the solution of the Cauchy problem for certain quasilinear equations for large time (Russian). Mat. Sb.,51 (1960), 191–216.

    MathSciNet  Google Scholar 

  3. Ya. I. Kanel’, On a model system of equations of one-dimensional gas motion (Russian). Differencial’nye Uravnenija,4 (1968), 374–380.

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. P.D. Lax, Hyperbolic systems of conservation laws II. Comm. Pure. Appl. Math.,10 (1957), 537–566.

    Article  MATH  MathSciNet  Google Scholar 

  6. J.T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws. Memoirs AMS,328 (1985), 1–108.

    Google Scholar 

  7. T.-P. Liu, A. Matsumura and K. Nishihara, Behavior of solutions for the Burgers equations with boundary corresponding to rarefaction waves. SIAM J. Math. Anal.,29(2) (1998), 293–308.

    Article  MATH  MathSciNet  Google Scholar 

  8. T.P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect. J. Differential Equations,133 (1997), 296–320.

    Article  MATH  MathSciNet  Google Scholar 

  9. T.-P. Liu and S.-H. Yu, Propagation of a stationary shock layer in the presence of a boundary. Arch. Rat. Mech. Anal.,139 (1997), 57–82.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Matsumura and K. Nishihara, Asymptotic stability of the traveling waves for scalar viscous conservation laws with non-convex nonlinearity. Comm. Math. Phys.,165 (1994), 83–96.

    Article  MATH  MathSciNet  Google Scholar 

  16. K. Nishihara, A note on the stability of traveling wave solution of Burgers’ equation. Japan J. Appl. Math.,2 (1985), 27–35.

    Article  MATH  MathSciNet  Google Scholar 

  17. A.I. Vol’pert and S.I. Hujaev, On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR-SB.,16 (1972), 517–544.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Guangxi University, 530004, Nanning, P.R. China

    Tao Pan

  2. Department of Mathematics, Jinan University, 510632, Guangzhou, P.R. China

    Hongxia Liu

  3. School of Political Science and Economics, Waseda University, 169-50, Tokyo, Japan

    Kenji Nishihara

Authors
  1. Tao Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongxia Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kenji Nishihara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Pan.

Additional information

To complete this work the first and second authors were supported in part by the Guangxi Education Committee Foundation GJK1998-69 and the KD Science Foundation of the State Council Office of Overseas Chinese Affairs. Research of the third author was supported in part by Waseda University Grant for Special Research Project 97A-242.

About this article

Cite this article

Pan, T., Liu, H. & Nishihara, K. Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary. Japan J. Indust. Appl. Math. 16, 431–441 (1999). https://doi.org/10.1007/BF03167367

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167367

Key words

Search

Navigation