The Boltzmann equation

An abstract form of the spatially non-homogeneous Boltzmann equation is derived which includes the usual, more concrete form for any kind of potential, hard or soft, with finite cutoff. It is assumed that the corresponding gas is confined to a bounded domain by some sort of reflection law. The problem then considered is the corresponding initial-boundary value problem, locally in time.Two proofs of existence are given. Both are constructive, and the first, at least, provides two sequences, one converging to the solution from above, the other from below, thus producing, at the same time as existence, approximations to the solution and error bounds for the approximation.The solution is found within a space of functions bounded by a multiple of a Maxwellian, and, in this space, uniqueness is also proved.Research supported, in part, by the National Research Council of Canada (NRC A8560)