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Theory of the Boltzmann equation

Conference paper

Abstract

The purpose of this article is to bring up to date earlier surveys [1, 2] of the state of the art in the theory of the Boltzmann equation for a dilute monatomic gas. Historically, the principal methods of attack have been connected with normal solutions and transport coefficients (Hilbbrt and Chapman-Enskog), polynomial and other more-or-less arbitrary approximations to the distribution function, integral iterations, and methods specific to models of the Boltzmann equation (e.g. Fourier transform). Each method has seen decisive advances in recent years. These advances are characterized by increased sophistication in mathematical techniques, improved understanding of the significance of older approximate results, and a general trend away from ad hoc and toward more precise mathematical procedures. In the singular limit of small mean free path, the traditional Hilbert and Chapman-Enskog expansions have been shown to be asymptotic to true solutions of the Boltzmann equation, but only when the variables are appropriately interpreted. At the opposite extreme of large mean free path where the behavior is again nonuniform, the precise mathematical singularities have been exposed. Over the whole range of linear problems the presence of a continuous spectrum in both the collision operator and the streaming operator points to the inadequacy of traditional exponential “normal mode” expansions in both initial and boundary value problems. Result in all the regimes have been tied together by the overall qualitative understanding given by a more comprehensive existence theory which, for the first time, is broad enough to encompass the transition to macroscopic continuum flow. Another attack, of very recent appearance, is direct numerical computation; this method is still largely exploratory.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1966

Authors and Affiliations

  • H. Grad
    • 1
  1. 1.New York UniversityUSA

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