Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Linearization for the Boltzmann equation

Author: F. Alberto Grünbaum
Journal: Trans. Amer. Math. Soc. 165 (1972), 425-449
MSC: Primary 82.45
MathSciNet review: 0295718
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we compare the nonlinear Boltzmann equation appearing in the kinetic theory of gases, with its linearized version. We exhibit an intertwining operator for the two semigroups involved. We do not assume from the reader any familiarity with Boltzmann's equation but rather start from scratch.

References [Enhancements On Off] (What's this?)

  • [1] T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler. 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957 (French). MR 0098477
  • [2] J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
  • [3] Harold Grad, Principles of the kinetic theory of gases, Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958, pp. 205–294. MR 0135535
  • [4] Harold Grad, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Sympos. Appl. Math., Vol. XVII, Amer. Math. Soc., Providence, R.I., 1965, pp. 154–183. MR 0184507
  • [5] Wolfgang Hahn, Stability of motion, Translated from the German manuscript by Arne P. Baartz. Die Grundlehren der mathematischen Wissenschaften, Band 138, Springer-Verlag New York, Inc., New York, 1967. MR 0223668
  • [6] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
  • [7] Kerson Huang, Statistical mechanics, John Wiley & Sons, Inc., New York-London, 1963. MR 0154659
  • [8] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press, Berkeley and Los Angeles, 1956, pp. 171–197. MR 0084985
  • [9] H. P. McKean Jr., Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas, Arch. Rational Mech. Anal. 21 (1966), 343–367. MR 0214112,
  • [10] H. Poincaré, Sur les propiétés des fonctions définies par les équations aux différences partielles. Vol. 1, Gauthier-Villars, Paris, 1929.
  • [11] A. Ja. Povzner, On the Boltzmann equation in the kinetic theory of gases, Mat. Sb. (N.S.) 58 (100) (1962), 65–86 (Russian). MR 0142362
  • [12] G. E. Uhlenbeck and G. W. Ford, Lectures in statistical mechanics, With an appendix on quantum statistics of interacting particles by E. W. Montroll. Lectures in Applied Mathematics (Proceedings of the Summer Seminar, Boulder, Colorado, vol. 1960, American Mathematical Society, Providence, R.I., 1963. MR 0151255
  • [13] E. Wild, On Boltzmann’s equation in the kinetic theory of gases, Proc. Cambridge Philos. Soc. 47 (1951), 602–609. MR 0042999

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 82.45

Retrieve articles in all journals with MSC: 82.45

Additional Information

Keywords: Boltzmann equation, spatially homogeneous case, nonlinear equation, linearized version, intertwining operator, eigenvalue inequalities
Article copyright: © Copyright 1972 American Mathematical Society