Theory of the Boltzmann Equation #225494

We present a survey of recent developments In the theory of the Boltzmann equation for a dilute monatomic gas. 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. Results 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.