We focus on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming Sn1 integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel-Shinbrot iteration technique to present elementary proofs of existence when the initial data is near vacuum and near a local Maxwellian. In the latter case we can also allow initial data with infinite mass. We study the propagation of regularity using a recent estimate for the gain operator, recently developed in collaboration with R. Alonso and E. Carneiro, that permits to study such propagation without additional conditions on the collision kernel. Finally, an L^p-stability result (with 1 < p < 1) is presented for this case assuming the aforementioned condition.
This is work in collaboration with Ricardo Alonso.