Advertisement

The Boltzmann Equation: Some Mathematical Aspects

  • C. Cercignani
Part of the International Centre for Mechanical Sciences book series (CISM, volume 293)

Abstract

After recalling the basic facts concerning the Boltzmann equation, a few recent developments are surveyed. Among these the trend to equilibrium, the H-theorem for polyatomic molecules, exact solutions in the space homogenous case and existence of affine homoenergetic flows.

Keywords

Boltzmann Equation Moment Equation Collision Operator Mathematical Aspect Collision Term 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1).
    C. CERCIGNANI - Transport Theory and Statistical Physics, 2, 211 (1982)CrossRefMathSciNetGoogle Scholar
  2. 2).
    C. CERCIGNANI - Theory and Application of the Boltzmann Equation, Scottish Academic Press, Edinburgh, and Elsevier, New York (1975)Google Scholar
  3. 3).
    M. AIZENMAN - Duke Math. J., 45, 809 (1978)CrossRefMATHMathSciNetGoogle Scholar
  4. 4).
    H. GRAD - In Handbuch der Physik, Vol. XII, Springer, Berlin (1958)Google Scholar
  5. 5).
    C. CERCIGNANI - Mathematical Methods in Kinetic Theory, Plenum Press, New York, and Mc Millan, London (1969)CrossRefMATHGoogle Scholar
  6. 6).
    O. LANFORD - In Springer Lecture Notes in Physics, 38, 1, ed. J. Moser, Springer, Berlin (1975)Google Scholar
  7. 7).
    H. SPOHN - In “Kinetic Theories and Boltzmann Equation”, ed. C. Cercignani, LNM 1948, Springer (1984).Google Scholar
  8. 8).
    C. CERCIGNANI - Transport Theory Statistical Physics, 2 (1972) 27.CrossRefMathSciNetGoogle Scholar
  9. 9).
    C. TRUESDELL and R.G. MUNCASTER - Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas ( Academic Press, New York, 1980 )Google Scholar
  10. 10).
    C. CERCIGNANI and M. LAMPIS - Transport Theory Statistical Physics, 1 (1971) 101CrossRefMATHMathSciNetGoogle Scholar
  11. 11).
    J. DARROZES and J. GUIRAUD - Compt. Rend. Acad. Sci. (Paris) A262 (1966) 1368Google Scholar
  12. 12).
    J. SCHNUTE - Canad. Jour. Math., 27, 1271, 1975CrossRefMathSciNetGoogle Scholar
  13. 13).
    L. BOLTZMANN - Sitzungsberichte der Akademie der Wissenschaften, Wien, 66, 275Google Scholar
  14. L. BOLTZMANN - Wissenschaftliche Abhandlungen, 1, 316, 1872Google Scholar
  15. 14).
    L. BOLTZMANN - Sitzungsberichte der Akademie der Wissenschaften, Wien, 72, 427; WissenschaftlicheGoogle Scholar
  16. 15).
    H.A. LORENZ–Wien Ber. 95–115 (1887) {also in Collected Papers, Martinus Nijhoff, éd., Vol. 6 p. 74}Google Scholar
  17. 16).
    S. CHAPMAN and T. G. COWLING - The Mathematical Theory of Non-Uniform gases ( Cambridge University Press. Cambridge, 1960 )MATHGoogle Scholar
  18. 36).
    J.C. MAXWELL - “Scientific Papers2, 26 and 2, 680, Dover, New York (1965)Google Scholar
  19. 37).
    H. GRAD - Comm. Pure Appl. Math., 3, 331 (1949)CrossRefMathSciNetGoogle Scholar
  20. 38).
    E. IKENBERRY and C. TRUESDELL - J. Rat. Mech. and Analysis, 5, 1 (1956)MATHMathSciNetGoogle Scholar
  21. 39).
    C. TRUESDELL - J. Rat. Mech. and Analysis, 5, 55 (1958)MathSciNetGoogle Scholar
  22. 40).
    M. KROOK and T.T. Wu - Phys. Rev. Lett. 36, 1107 (1976)CrossRefGoogle Scholar
  23. M. KROOK and T.T. Wu - Phys. Fluids 20, 1589 (1977)CrossRefMATHGoogle Scholar
  24. 41).
    A.V. BOBYLEV - Sov. Phys. Doklady, 20, 820 and 822 (1976)Google Scholar
  25. A.V. BOBYLEV - Sov. Phys. Doklady, 21, 632 (1977)Google Scholar
  26. 42).
    R. KRUPP - Private comunication (1967)Google Scholar
  27. 43).
    M. H. ERNST - Phys. Rep. 78, 1 (1981)CrossRefMathSciNetGoogle Scholar
  28. 44).
    M. H. ERNST - In Nonequilibrium Phenomena I - The Boltzmann Equation, J.L. LEBOWITZ and E. W. MONTROLL, eds. North Holland, Amsterdam (1983)Google Scholar
  29. 45).
    C. TREUSDELL–“On the pressures and the flux of energy in a gas according to Maxwell’s Kinetic theory, II”, Journal of Rational Mechanics and Analysis, 5, 55–128 (1956)Google Scholar
  30. 46).
    V. S. GALKIN - Prikladnaya Matematika i Mechanika, 20, 445–446 (1956) (In Russian)Google Scholar
  31. 47).
    V. S. GALKIN–“On a class of solutions of Grad’s moment equations”, PMM 22, 532–536 (1958)MATHMathSciNetGoogle Scholar
  32. 48).
    V. S. GALKIN–“One-dimensional unsteady solution of the equation for the kinetic moments of a monatomic gas”, PMM 28, 226–229 (1964)MathSciNetGoogle Scholar
  33. 49).
    V. S. GALKIN–“Exact solutions of the kinetic moment equations of a mixture of monatomic gases”, Fluid Dynamics, 1, 29–34 (1966)CrossRefGoogle Scholar
  34. 50).
    C. TRUESDELL and R. G. MUNCASTER - Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatómic. Gas, New York, Academic Press (1980)Google Scholar
  35. 51).
    C. CERCIGNANI- Submitted to Archive for Rational Mechanics and Analy sis (1986)Google Scholar
  36. 17).
    L. BOLTZMANN - Vorlesungen über Gastheorie, J. A. Barth, ed. (Leipzig, 1898) Vol. 2, Chap VII {also in Lectures on Gas Theory, transl. S.G. Brush (Berkeley, 1964)}Google Scholar
  37. 18).
    R.C. TOLMAN - The Principles of Statistical Mechanics (Oxford Unive rsity Press, London, 1938 )Google Scholar
  38. 19).
    G. H. BRYAN - Brit. Assoc. Reports, p. 64 (1894)Google Scholar
  39. 20).
    F. B. PIDDUCK - Proc. R. Soc. London Ser. A 101 (1922)Google Scholar
  40. 21).
    L. BOLTZMANN–Wien Ber. 95–153 (1887) {also in Wiss. Abh. 3–272}Google Scholar
  41. 22).
    G. E. UHLENBECK - In The Boltzmann Equation - Theory and Applications, E. G. D. COHEN and W. THIRRING, eds. Acta Phys. Austr., Suppl X, 107 ( Springer-Verlag, Vienna, 1973 )Google Scholar
  42. 23).
    L. WALDMANN - Z. Naturforsch, 12A - 660 (1957)MATHGoogle Scholar
  43. 24).
    L. WALDMANN - Z. Naturforsch, 13A - 609 (1958)MATHMathSciNetGoogle Scholar
  44. 25).
    R.F. SNIDER–J. Chem. Phys. 32–1051 (1960)CrossRefMathSciNetGoogle Scholar
  45. 26).
    E. C. G. STUECKELBERG–Helv.Pys. Acta 25–577 (1952)MATHMathSciNetGoogle Scholar
  46. 27).
    L. WALDMANN - In Handbuch der Physik, S. FLUGGE ed. ( Springer-Verlag, Berlin, 1958 ), Vol. 12, p. 484Google Scholar
  47. 28).
    C. CERCIGNANI and M. LAMPIS - J. Stat. Phys., 26, 795 (1981)CrossRefMathSciNetGoogle Scholar
  48. 29).
    L. WALDMANN - In the Boltzmann Equation - Theory and Applications, E. G. D. COHEN and W. THIRRING, eds. Acta Phys. Austr., Suppl. X, 107, ( Springer-Verlag, Vienna 1973 )Google Scholar
  49. 30).
    E. WILD - Proc. Camb. Phil. Soc., 47, 602, 1950CrossRefMathSciNetGoogle Scholar
  50. 31).
    D. MORGENSTERN - J. Rat. Mech. Anal. 4, 533, 1955MATHMathSciNetGoogle Scholar
  51. 32).
    L. ARKERYD - Arkive Rat. Mech Anal; 45, 1 and 17, 1972Google Scholar
  52. 33).
    M. PITTERI - Acc. Naz Lincei (Roma). Sez. VIII, 67, 248, 1979MathSciNetGoogle Scholar
  53. 34).
    C. CERCIGNANI - Arch. Mech. 34, 231 (1982)MATHMathSciNetGoogle Scholar
  54. 35).
    T. ELMROTH - Ph. D. Thesis, Göteborg (1984)Google Scholar
  55. 52).
    A. YA. POVZNER - “The Boltzmann equation in the kinetic theory of gases”, American Mathematical Society Translations 47, (2) 193–216Google Scholar

Copyright information

© Springer-Verlag Wien 1988

Authors and Affiliations

  • C. Cercignani
    • 1
  1. 1.Politecnico di MilanoMilanItaly

Personalised recommendations