Skip to main content
SpringerLink
Log in

Lyapunov Functionals and L 1-Stability for Discrete Velocity Boltzmann Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We devise Lyapunov functionals and prove uniform L 1 stability for one-dimensional semilinear hyperbolic systems with quadratic nonlinear source terms. These systems encompass a class of discrete velocity models for the Boltzmann equation. The Lyapunov functional is equivalent to the L 1 distance between two weak solutions and non-increasing in time. They result from computations of two point interactions in the phase space. For certain models with only transversal collisional terms there exist generalizations for three and multi-point interactions.

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

Access this article

Log in via an institution

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

    MathSciNet  MATH  Google Scholar 

  2. Beale, J.-T.: Large-time behavior of discrete velocity Boltzmann equations. Commun. Math. Phys. 106 (4), 659–678 (1986)

    MATH  Google Scholar 

  3. Bony, J.-M.: Solutions globales bornées pour les modèles discrets de l' équation de Boltzmann, en dimension 1 d'espace. In: Journées "Équations aux derivées partielles" (Saint Jean de Monts, 1987), Exp. No. XVI. 10. pp. Palaiseau: École Polytech, 1987

  4. Bony, J.-M.: Existence globale et diffusion en théorie cinétique discrete. In: Advances in kinetic theory and continuum mechanics. R. Gatignol and Soubbaramayer, eds, Berlin-Heidelberg-New York: Springer-Verlag, 1991, pp. 81–90

  5. Bony, J.-M.: Problème de Cauchy et diffusion à données petites pour les modèles discrets de la cinétique des gaz. In: Journées ``Équations aux derivées partielles'' (Saint Jean de Monts, 1990), Exp. No. I. 12. pp. Palaiseau: École Polytech, 1990

  6. Bony, J.-M.: Existence globale à données de Cauchy petites pour les modèles discrets de l'équation de Boltzmann. Comm. Partial Differential Equations 16, 533–545 (1991)

    MathSciNet  MATH  Google Scholar 

  7. Bony, J.-M.: Existence globale et diffusion pour les modèles discrets de la cinétique des gaz. In: First European Congress of Mathematics, Vol. I (Paris, 1992), Progr. Math. 119, Basel: Birkäuser, 1994, pp. 391–410

  8. Bose, C., Grzegorrczyk, P., Illner, R.: Asymptotic behavior of one-dimensional discrete velocity models in a slab. Arch. Rational Mech. Anal. 127, 337–360 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Bressan, A., Liu, T.-P., Yang, T.: L 1 stability estimates for n × n conservation laws. Arch. Rational Mech. Anal. 149, 1–22 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. Cabannes, H.: Solution globale du problème de Cauchy en théorie cinétique discrète. J. Mécanique 17, 1–22 (1978)

    MATH  Google Scholar 

  12. Cercignani, C.: A remarkable estimate for the solutions of the Boltzmann equation. Appl. Math. Lett. 5, 59–62 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cercignani, C.: Weak solutions of the Boltzmann equation and energy conservation. Appl. Math. Lett. 8, 53–59 (1995); See also: Errata. Appl. Math. Lett. 8(5), 95–99 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cercignani, C., Illiner, R.: Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions. Arch. Rat. Mech. Anal. 134, 1–16 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Gatignol, R.: Théorie Cinétique des Gaz a Répartition Discrète de Vitesses. Lecture Notes in Physics, 36, Berlin-New York: Springer-Verlag, 1975

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

    MATH  Google Scholar 

  17. Hu, J., LeFloch, Ph.: L 1 continuous dependence property for systems of conservation laws. Arch. Ration. Mech. Anal. 151, 45–93 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Illner, R.: Global existence results for discrete velocity models of the Boltzmann equation. J. Meca. Th. Appl. 1, 611–622 (1982)

    MathSciNet  MATH  Google Scholar 

  19. Platkowski, T., Illner, R.: Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory. SIAM Review 30, 213–255 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Katsoulakis, M., Tzavaras, A.: Contractive relaxation systems and the scalar multidimensional conservation law. Comm. Partial Differential Equations 22, 195–233 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Kawashima, S.: Global solution of the initial value problem for a discrete velocity model of the Boltzmann equation. Proc. Japan Acad. 57, 19–24 (1981)

    MathSciNet  MATH  Google Scholar 

  22. Liu, T.-P., Yang, T.: Well posedness theory for hyperbolic conservation laws. Comm. Pure Appl. Math. 52, 1553–1586 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nishida, T., Mimura, M.: On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas. Proc. Japan. Acad. 50, 812–817 (1974)

    MATH  Google Scholar 

  24. Schatzman, M.: Continuous Glimm functionals and uniqueness of solutions of the Riemann problem. Indiana Univ. Math. J. 34, 533–589 (1985)

    MathSciNet  MATH  Google Scholar 

  25. Tartar, L.: Existence globale pour un système hyperbolique semi linéaire de la théorie cinétique des gaz. In: Séminaire Goulaouic-Schwartz (1975/1976), Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 1, 11 pp. Centre Math. Palaiseau: École Polytech, 1976

    Google Scholar 

  26. Tartar, L.: Some existence theorems for semilinear hyperbolic systems in one space variable. MRC Technical Summary Report, No. 2164. University of Wisconsin-Madison (1980)

  27. Tartar, L. (1987): Oscillations and asymptotic behaviour for two semilinear hyperbolic systems. In: Dynamics of infinite-dimensional systems. NATO Adv. Sci. Inst. Ser. F, 37, Berlin: Springer, 1981, pp. 341–356

  28. Tzavaras, A.: On the mathematical theory of fluid dynamic limits to conservation laws. In: Advances in Mathematical Fluid Mechanics. J. Malek, J. Necas, M. Rokyta, eds.; New York: Springer, 2000, pp. 192-222

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA

    Seung-Yeal Ha & Athanasios E. Tzavaras

  2. Institute for Applied and Computational Mathematics, FORTH, 1527, 71110, Heraklion, Crete, Greece

    Athanasios E. Tzavaras

Authors
  1. Seung-Yeal Ha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Athanasios E. Tzavaras
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seung-Yeal Ha.

Additional information

J.L. Lebowitz

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ha, SY., Tzavaras, A. Lyapunov Functionals and L 1-Stability for Discrete Velocity Boltzmann Equations. Commun. Math. Phys. 239, 65–92 (2003). https://doi.org/10.1007/s00220-003-0866-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-0866-9

Keywords

Search

Navigation