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On the asymptotic theory of the Boltzmann and Enskog equations a rigorous H-theorem for the Enskog equation

Part of the Lecture Notes in Mathematics book series (LNM,volume 1460)

Abstract

This paper developes the mathematical theory of the asymptotic equivalence, initiated by the authors in a previous paper [3], between the Boltzmann and the Enskog equations referred to the solutions to the initial value problem when the radius of the hard spheres in the Enskog equation tends to zero. This paper deals with an H-theorem for the Enskog equation and proves an asymptotic equivalence result which states that the Liapunov functional proposed by Polewczak [13], referred to the solution of the initial value problem for the Enskog equation, is monotone decreasing in time and tends, when the radius of the spheres goes to zero, to the H function referred to the Boltzmann equation.

Keywords

  • Boltzmann Equation
  • Hard Sphere
  • Asymptotic Theory
  • Knudsen Number
  • Pair Correlation Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On leave from Depart. Mathematics, University of Warsaw, Poland

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References

  1. Arkeryd L. and Cercignani C., To be published.

    Google Scholar 

  2. Bellomo N. and Lachowicz M., Int. J. Modern Phys. B., 1, (1987), 1193–1206.

    CrossRef  ADS  MathSciNet  Google Scholar 

  3. Bellomo N. and Lachowicz M., J. Statist. Phys, 51, (1988), 233–247.

    CrossRef  ADS  MathSciNet  Google Scholar 

  4. Bellomo N., Palczewski A. and Toscani G., Mathematical Topics in Nonlinear Kinetic Theory, World Scientific, London, Singapore, (1988).

    MATH  Google Scholar 

  5. Di Perna R. and Lions P.L., Comp. Rend. Acad. Sci. Paris, t.306, I, (1988), 343–346.

    Google Scholar 

  6. Di Perna R. and Lions P.L., Internal Report, Univ. de Paris Dauphine, n. 8903, (1989).

    Google Scholar 

  7. Enskog D. Svenska Akad., 4, (1921), 3–44.

    Google Scholar 

  8. Lachowicz M., Arch. Rational. Mech. Anal., 101, (1988), 179–194.

    CrossRef  ADS  MathSciNet  Google Scholar 

  9. Lachowicz M., Ann.Mat. Pura Appl., To appear.

    Google Scholar 

  10. Lebowitz J., Percus J. and Sykes J., Physical Review, 188, (1968), 487–507.

    CrossRef  ADS  Google Scholar 

  11. Marechal M., Blawzdziewicz J. and Piasecki J., Phys. Rew. Lett., 52, (1984), 1169–1174.

    CrossRef  ADS  Google Scholar 

  12. Polewczak J., SIAM J. Appl. Math., To appear.

    Google Scholar 

  13. Polewczak J., J. Statist. Phys, To Appear.

    Google Scholar 

  14. Resibois P. and De Leener M., Classical Kinetic Theory of Fluids, Wiley, London, (1977).

    MATH  Google Scholar 

  15. Resibois P., J. Stat. Phys., 19, (1978), 593–609.

    CrossRef  ADS  Google Scholar 

  16. Toscani G., Arch. Rational. Mech. Anal., 100, (1987), 1–12.

    CrossRef  ADS  MathSciNet  Google Scholar 

  17. Van Beijeren H. and Ernst M. H., Physica, 68, (1973), 437–456.

    CrossRef  ADS  Google Scholar 

  18. Van Beijeren H., Phys. Rew. Lett., 51 (1983), 1503–1506.

    CrossRef  ADS  Google Scholar 

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© 1991 Springer-Verlag

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Bellomo, N., Lachowicz, M. (1991). On the asymptotic theory of the Boltzmann and Enskog equations a rigorous H-theorem for the Enskog equation. In: Toscani, G., Boffi, V., Rionero, S. (eds) Mathematical Aspects of Fluid and Plasma Dynamics. Lecture Notes in Mathematics, vol 1460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091358

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  • DOI: https://doi.org/10.1007/BFb0091358

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53545-4

  • Online ISBN: 978-3-540-46779-3

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