Skip to main content
Log in

Asymptotics toward rarefaction wave of solutions of the Broadwell model of a discrete velocity gas

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

Abstract

This paper studies the asymptotic behavior toward rarefaction wave for solutions of the initial value problem to the one-dimensional Broadwell model of the Boltzmann equation. When the Riemann problem for the Euler equation, derived from the Chapman-Enskog expansion, admits the solution of weak rarefaction wave, we also call the corresponding local Maxwellian of the original Broadwell model “rarefaction wave”. Then if the initial data are suitably close to the rarefaction wave at the initial time, the solution is proven to tend toward the rarefaction wave as time goes to infinity. The proof 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

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. J. T. Beale, Large-time behavior of the Broadwell model of a discrete velocity gas. Comm. Math. Phys.,102 (1985), 217–235.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. E. Broadwell, Shock structure in a simple discrete velocity gas. Phys. Fluids,7 (1964), 1243–1247.

    Article  MATH  Google Scholar 

  3. R. E. Caflisch, Navier-Stokes and Boltzmann shock profiles for a model of gas dynamics. Comm. Pure Appl. Math.,32 (1979), 521–554.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Inoue and T. Nishida, On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas. Appl. Math. Optim.,3 (1976), 27–49.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Kawashima, The asymptotic equivalence of the Broadwell model equation and its Navier-Stokes model equation. Japan. J. Math.,7 (1981) 1–43.

    MATH  MathSciNet  Google Scholar 

  6. S. Kawashima, Global existence and stability of solutions for discrete velocity models of the Boltzmann equation. Lecture Notes in Numer. Appl. Anal. 6, Recent Topics in Nonlinear PDE, Hiroshima 1983, 1983, 59–85.

  7. S. Kawashima, Large-time behavior of solutions of the discrete Boltzmann equation. To appear.

  8. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Matsumura and K. Nishihara, Asymptotics 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 

Download references

Author information

Authors and Affiliations

Authors

About this article

Cite this article

Matsumura, A. Asymptotics toward rarefaction wave of solutions of the Broadwell model of a discrete velocity gas. Japan J. Appl. Math. 4, 489–502 (1987). https://doi.org/10.1007/BF03167816

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167816

Key words

Navigation