WAVE PROPAGATION AND OTHER SPECTRAL PROBLEMS IN KINETIC THEORY

A rigorous discussion of the spectrum of the linear Boltzmann equation and kinetic models is presented. Particular attention is given to plane-wave propagation for a general class of kinetic models. These models in general have a velocity-dependent collision frequency v(ξ). The main results of this paper concern the relationship between the complex wavenumber and complex frequency for a plane wave. It is shown that the question of analyticity of this relation is reduced to considering v in the neighborhood of infinity. Specifically, if lim v/ξ = 0, ξ→∞ the relationship is not analytic. Otherwise, analyticity is obtained. (Although not specifically considered here, analyticity is closely connected to the convergence of the Chapman-Enskog procedure.) In a general discussion it is shown how the question of analyticity is closely connected with (i) the continuous spectrum of the underlying operator, (ii) the behavior of solutions at large distances from boundaries, and (iii) the nature of the cutoff in an intermolecular interaction.