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The Boltzmann equation and its properties

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1048)

Keywords

  • Boltzmann Equation
  • Mild Solution
  • Specular Reflection
  • Iteration Scheme
  • Local Existence

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References and Footnotes

  1. We adopt the notation and techniques described by James J. Duderstadt and William K. Martin ‘Transport Theory" (John Wiley, New York, 1979). Chapter 3. This textbook contains a number of references to the earlier literature.

    MATH  Google Scholar 

  2. The treatment follows that of Kerson Huang "Statistical Mechanics" (John Wiley, New York, 1963) Chapter 4.

    Google Scholar 

  3. Joel L. Lebowitz and Peter G. Bergmann, Ann. Phys. (N.Y.) 1, 1(1957).

    CrossRef  ADS  MathSciNet  Google Scholar 

  4. H. Grad in Handbuch der Physik, Vol. XII. "Thermodynamics of Gases" (Springer-Verlag, Berlin, 1968).

    Google Scholar 

  5. Sometimes, in order to compress notation, we shall use the symbol (\(\vec w\)) to indicate the molecule whose velocity is \(\vec w\).

    Google Scholar 

  6. See, for example, P. F. Zweifel "Reactor Physics" (McGraw-Hill, New York, 1973) Appendix E.

    Google Scholar 

  7. H. Goldstein. "Classical Mechanics" (Addison-Wesley, Cambridge, Mass., 1953) page 82.

    Google Scholar 

  8. Michael Reed and Barry Simon "Methods of Modern Mathematical Physics-II. Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975) Sec. X. 13.

    MATH  Google Scholar 

  9. Walter Rudin. "Real and Complex Analysis" (McGraw-Hill, New York, 1966) page 21.

    MATH  Google Scholar 

  10. T. Carleman. "Problemès mathématiques dans la théorie cinétique des gaz," (Almqvist and Wiksells, Uppsala. 1957).

    MATH  Google Scholar 

  11. E. Wild, Proc. Camb. Phil. Soc. 47, 602 (1951).

    CrossRef  ADS  MathSciNet  Google Scholar 

  12. D. Morgenstern, Proc. Nat. Acad. Sci. U.S.A. 40, 719 (1954).

    CrossRef  ADS  MathSciNet  Google Scholar 

  13. Lief Arkeryd Arch. Rat. Mech. and Anal. 95, 1 (1972)

    ADS  MathSciNet  Google Scholar 

  14. 17 (1972). c.f. also

    Google Scholar 

  15. "Intermolecular Forces of Infinite Range and the Boltzmann Equation," Chalmers University of Technology (Sweden) preprint (1970).

    Google Scholar 

  16. A. Ja. Povzner, Mat. Sbornik 58, 65 (1962).

    MathSciNet  Google Scholar 

  17. G. Di Blasio, Boll. U.M.I. 8, 127 (1973); Comm. Math. Phys. 38, 331 (1974).

    MathSciNet  Google Scholar 

  18. Shmuel Kaniel and Marvin Shinbrot, Comm. Math. Phys. 58, 65 (1978).

    CrossRef  ADS  MathSciNet  Google Scholar 

  19. Alexander Glikson, Arch. Rat. Mech. Anal. 45, 35 (1972); Bull. Australian Math. Soc. 16, 321 (1977).

    CrossRef  MathSciNet  Google Scholar 

  20. D. Morgenstern, J. Rat. Anal. 4, 533 (1955).

    MathSciNet  Google Scholar 

  21. Barry Simon "The P(φ)2 Euclidean (Quantum) Field Theory" (Princeton Univ. Press, Princeton, N.J., 1974).

    Google Scholar 

  22. Michael Reed and Barry Simon "Methods of Modern Mathematical Physics—I; Functional Analysis" (Academic Press, New York, 1972) p. 151.

    MATH  Google Scholar 

  23. The proof of Lemma 2 requires that R(t1)R(t2) = R(t2)R(t1), otherwise, the "time-ordered" exponential must be used. cf. Chapter VI for a situation in which this is necessary.

    Google Scholar 

  24. H. Grad in "Applications of Nonlinear Partial Differential Equations in Mathematical Physics" (Amer. Math. Soc., Providence, R.I., 1965) p. 154.

    Google Scholar 

  25. cf. Ref. 5, p. 237.

    Google Scholar 

  26. cf. Ref. 24, Appendix.

    Google Scholar 

  27. cf. Ref. 2, p. 131.

    Google Scholar 

  28. H. Grad, Comm. Pure Appl. Math. XVIII, 315 (1965).

    Google Scholar 

  29. H. Grad, Phys. Fluids 6, 147 (1963).

    CrossRef  ADS  MathSciNet  Google Scholar 

  30. Y. Shizuta and K. Asano, Proc. Japan Acad. 53, 3 (1977).

    CrossRef  MathSciNet  Google Scholar 

  31. T. Kato "Perturbation Theory for Linear Operators" (Springer-Verlag, New York, 1966) Chapter IX.

    CrossRef  MATH  Google Scholar 

  32. T. Nishida and K. Imai, Publ. RIMS Kyoto Univ. 229 (1976).

    Google Scholar 

  33. J. P. Guraud, Coll. Int. C.N.R.S. No. 236.

    Google Scholar 

  34. Y. Shizuta, "On the Classical Solutions of the Boltzmann Equation." Preprint.

    Google Scholar 

  35. S. Ukai, Proc. Japan Acad. 50, 179 (1974); C. R. Acad. Sci. Paris 282, A-317 (1976).

    CrossRef  MathSciNet  Google Scholar 

  36. C. Cercignani, W. Greenberg and P. F. Zweifel, J. Stat. Phys. 20, 449 (1979).

    CrossRef  ADS  MathSciNet  Google Scholar 

  37. Herbert Spohn, J. Stat. Phys. 20, 463 (1979).

    CrossRef  ADS  MathSciNet  Google Scholar 

  38. W. Greenberg, J. Voigt and P. F. Zweifel, J. Stat. Phys. 21, 649 (1979).

    CrossRef  ADS  MathSciNet  Google Scholar 

  39. cf. Sec. 12, Lemma 2. Morgenstern's operator R(t;f) is a multiplicative operator, and thus the evolution is given simply by \(\int\limits_0^t {R(t;f(\tau ))d\tau }\) In our case, T(n;t1,t2) is generated by A + v (n) and is hence a non-diagonal matrix, which is why we find it necessary to introduce "time-ordering." Somehow, Morgenstern has been able to circumvent this difficulty by working along the free trajectories.

    Google Scholar 

  40. See Ref. 31, pp. 487 ff.

    Google Scholar 

  41. J. P. Ginzburg, Am. Math. Soc. Translations, Series 2, 96, 189 (1970).

    Google Scholar 

  42. Ref. 21, Sec. VIII.8.

    Google Scholar 

  43. J. Voigt, T.T.S.P. 8, 17 (1979).

    MathSciNet  Google Scholar 

  44. M. Moreau, J. Math. hys. 19, 2494 (1978).

    ADS  MathSciNet  Google Scholar 

  45. J. Voigt "H-theorem for Boltzmann Type Equations." Preprint, Laboratory for Transport Theory and Mathematical Physics, Va. Polytech. Inst. (1979).

    Google Scholar 

  46. H. L. Royden "Real Analysis" (MacMillan, London, 1968) p. 110.

    Google Scholar 

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© 1984 Springer-Verlag

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Zweifel, P.F. (1984). The Boltzmann equation and its properties. In: Cercignani, C. (eds) Kinetic Theories and the Boltzmann Equation. Lecture Notes in Mathematics, vol 1048. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071879

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  • DOI: https://doi.org/10.1007/BFb0071879

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