Advertisement

The Boltzmann Equation

Chapter
Part of the Progress in Mathematical Physics book series (PMP, volume 41)

Keywords

Boltzmann Equation Hard Sphere Bulk Velocity Grad Limit Mann Equation 
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.
    L. Arkeryd, On the Boltzmann equation. Part I: Existence, Arch. Rat. Mech. Anal. 45, 1–16 (1972).zbMATHMathSciNetGoogle Scholar
  2. 2.
    L. Arkeryd, On the Boltzmann equation. Part II: The full initial value problem, Arch. Rat. Mech. Anal. 45, 17–34 (1972).zbMATHMathSciNetGoogle Scholar
  3. 3.
    L. Arkeryd and C. Cercignani, On a functional equation arising in the kinetic theory of gases, Rend. Mat. Acc. Lincei s.9, 11, 139–149 (1990).MathSciNetGoogle Scholar
  4. 4.
    L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Sitzungsberichte Akad. Wiss., Vienna, part II, 66, 275–370 (1872).zbMATHGoogle Scholar
  5. 5.
    T. Carleman, Sur la théorie de l’équation intégro-differentielle de Boltzmann, Acta Mathematica 60, 91–146 (1933).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    T. Carleman, Problémes Mathématiques dans la Théorie Cinétique des Gaz, Almqvist & Wiksell, Uppsala (1957).zbMATHGoogle Scholar
  7. 7.
    C. Cercignani, The Boltzmann equation and its applications, Springer, New York (1988).zbMATHGoogle Scholar
  8. 8.
    C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York (1969), Revised edition (1990).zbMATHGoogle Scholar
  9. 9.
    C. Cercignani, On the Boltzmann equation for rigid spheres, Transp. Theory Stat. Phys., 211–225 (1972).Google Scholar
  10. 10.
    C. Cercignani, Are there more than five linearly independent collision invariants for the Boltzmann equation?, J. Statistical Phys. 58, 817–824 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    C. Cercignani, Ludwig Boltzmann. The man who trusted atoms, Oxford University Press, Oxford (1998).zbMATHGoogle Scholar
  12. 12.
    S. Chapman, The kinetic theory of simple and composite gases: Viscosity, thermal conduction and diffusion, Proceedings of the Royal Society (London) A93, 1–20 (1916/17).Google Scholar
  13. 13.
    R. Di Perna and P. L. Lions, On the Cauchy problem for Boltzmann equations, Ann. of Math. 130, 321–366 (1989).zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Enskog, Kinetische Theorie der Vorgänge in mässig verdünnten Gasen, I.Allgemeiner Teil, Almqvist & Wiksell, Uppsala (1917).zbMATHGoogle Scholar
  15. 15.
    V. S. Galkin, Ob odnom resheniř kineticheskogo uravneniya, Prikladnaya Matematika i Mekhanika 20, 445–446 (1956) (in Russian).MathSciNetGoogle Scholar
  16. 16.
    V. S. Galkin, On a class of solutions of Grad’s moment equations, PMM 22, 532–536 (1958).zbMATHMathSciNetGoogle Scholar
  17. 17.
    H. Grad, On the kinetic theory of rarified gases, Comm. Pure Appl. Math. 2, 331–407 (1949).zbMATHMathSciNetGoogle Scholar
  18. 18.
    H. Grad, Asymptotic equivalence of the Navier-Stokes and non-linear Boltzmann equation, Proceedings of the American Mathematical Society Symposia on Applied Mathematics 17, 154–183 (1965).zbMATHMathSciNetGoogle Scholar
  19. 19.
    D. Hilbert, Begründung der kinetischen Gastheorie, Mathematische Annalen 72, 562–577 (1916/17).MathSciNetCrossRefGoogle Scholar
  20. 20.
    R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in a vacuum, Commun. Math. Phys. 105, 189–203 (1986).zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two-and three-dimensional rare gas in vacuum: Erratum and improved result, Comm. Math. Phys. 121, 143–146 (1989).zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    R. Illner and M. Shinbrot, The Boltzmann equation: Global existence for a rare gas in an infinite vacuum Comm. Math. Phys., 95, 217–226 (1984).zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    O. Lanford, III, The evolution of large classical systems, in Dynamical Systems, Theory and Applications, J. Moser, Ed., LNP 35, 1–111, Springer, Berlin (1975).Google Scholar
  24. 24.
    J. C. Maxwell, On the Dynamical Theory of Gases, Philosophical Transactions of the Royal Society of London 157, 49–88 (1867).Google Scholar
  25. 25.
    D. Morgenstern, General existence and uniqueness proof for spatially homogeneous solutions of the Maxwell-Boltzmann equation in the case of Maxwellian molecules, Proceedings of the National Academy of Sciences (U.S.A.) 40, 719–721 (1954).zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    T. Nishida and K. Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation, Publications of the Research Institute for Mathematical Sciences, Kyoto University 12, 229–239 (1977).MathSciNetGoogle Scholar
  27. 27.
    Y. Shizuta and K. Asano, Global solutions of the Boltzmann equation in a bounded convex domain, Proceedings of the Japan Academy 53, 3–5 (1974).MathSciNetGoogle Scholar
  28. 28.
    C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory, II, Jour. Rat. Mech. Anal. 5, 55–128 (1956).zbMATHMathSciNetGoogle Scholar
  29. 29.
    S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proceedings of the Japan Academy 50, 179–184 (1974).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag 2006

Personalised recommendations