Linearization for the Boltzmann equation

Grünbaum, F. Alberto

doi:10.1090/S0002-9947-1972-0295718-5

Linearization for the Boltzmann equation
HTML articles powered by AMS MathViewer

by F. Alberto Grünbaum PDF

Trans. Amer. Math. Soc. 165 (1972), 425-449 Request permission

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

T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler, vol. 2, Almqvist & Wiksells Boktryckeri AB, Uppsala, 1957 (French). MR 0098477
J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
Harold Grad, Principles of the kinetic theory of gases, Handbuch der Physik, Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. Herausgegeben von S. Flügge. MR 0135535
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
Wolfgang Hahn, Stability of motion, Die Grundlehren der mathematischen Wissenschaften, Band 138, Springer-Verlag New York, Inc., New York, 1967. Translated from the German manuscript by Arne P. Baartz. MR 0223668
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Kerson Huang, Statistical mechanics, John Wiley & Sons, Inc., New York-London, 1963. MR 0154659
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-Los Angeles, Calif., 1956, pp. 171–197. MR 0084985
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 214112, DOI 10.1007/BF00264463

Sur les propiétés des fonctions définies par les équations aux différences partielles

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
G. E. Uhlenbeck and G. W. Ford, Lectures in statistical mechanics, Lectures in Applied Mathematics (Proceedings of the Summer Seminar, Boulder, Colorado, vol. 1960, American Mathematical Society, Providence, R.I., 1963. With an appendix on quantum statistics of interacting particles by E. W. Montroll. MR 0151255
E. Wild, On Boltzmann’s equation in the kinetic theory of gases, Proc. Cambridge Philos. Soc. 47 (1951), 602–609. MR 42999, DOI 10.1017/s0305004100026992