Skip to main content
Log in

Decay of solutions for the equations of isothermal gas dynamics

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

Abstract

The equations of isothermal gas dynamics are expressed as a 2×2-system of genuinely nonlinear hyperbolic conservation laws which possesses a convex entropy. Existence of weak global solutions is known for even large initial data via Glimm’s difference scheme. We show that the total variation of such (large) solutions decays strongly to zero. The proof consists in showing that the total amount of wave interaction in the Glimm approximations is uniformly bounded.

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

Access this article

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. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves. Wiley-Interscience, New York, 1948.

    MATH  Google Scholar 

  2. R. J. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws. Indiana Univ. Math J.,24 (1975), 1047–1071.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. J. DiPerna, Decay of solutions of hyperbolic systems of conservation laws with a convex extension. Arch. Rational Mech. Anal.,64 (1977), 1–46.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension. Proc. Nat. Acad. Sci., U.S.A.,68 (1971), 1686–1688.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math.,18 (1965), 697–715.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Glimm and P. D. Lax, Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws. Amer. Math. Soc. Mem., No. 101, A.M.S., Providence, 1970.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. P. D. Lax, Shock waves and entropy. Contributions to Nonlinear Functional Analysis (ed. E. Zarantonello), Academic Press, New York, 1971, 603–634.

    Google Scholar 

  9. T. P. Liu, Large time behavior of initial and initial-boundary-value problems of general systems of hyperbolic conservation laws. Comm. Math. Phys.,55 (1977), 163–177.

    Article  MATH  MathSciNet  Google Scholar 

  10. T. P. Liu, Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Comm. Pure Appl. Math.,30 (1977), 585–610.

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. T. P. Liu, The deterministic version of the Glimm scheme. Comm. Math. Phys.,57 (1977), 135–148.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. P. Liu, Asymptotic behavior of solutions of general system of nonlinear hyperbolic conservation laws. Indiana Univ. Math. J.,27 (1978), 211–253.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Nishida, Global solution for an initial boundary value problem of a quasilinear hyperbolic system. Proc. Japan Acad.,44 (1968), 642–646.

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics. Publ. Math. No. 79-02. Univ. Paris-Sud, 1978.

  16. J. Smoller, Shock Waves and Reaction-Diffusion Equations. Springer Verlag, New York, 1983.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

About this article

Cite this article

Asakura, F. Decay of solutions for the equations of isothermal gas dynamics. Japan J. Indust. Appl. Math. 10, 133–164 (1993). https://doi.org/10.1007/BF03167207

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167207

Key words

Navigation