Formal and Asymptotic Solutions in Kinetic Theory

Formal solutions are found to the linearized Boltzmann equation for the initial-value problem. These are decomposed into an infinity of modes, which are orthogonal under a suitable inner product. All but five of these modes exhibit an exponential time decay. These five remaining modes form a generalization of hydrodynamics. Under two different asymptotic assumptions one finds quantitative solutions. If the characteristic wavelength of the initial disturbance is large compared to the mean free path, the solution appears as an expansion in their ratio. If the elapsed time is large compared to the mean free time, the solution may be represented as an expansion in their ratio. As a specific example of the theory, the fundamental solution of the one-dimensional shear-free initial value problem is computed. This appears as an infinity of diffusing modes, a subclass of which also propagate.