Sharp estimates to solutions of homogeneous Boltzmann type problems
We have recently derived sharper form of the Povzner Lemma which essentially provides control of the gain operator for classical bilineal collisional integral forms for hard spheres with rather general differential cross sections. This estimate provides the tool, combined with recent maximum principle for this Boltzmann type of equations and analysis of the Carleman representation of the gain operator, to obtain pointwise bounds to stationary smooth solutions of these type of Boltzmann equations by integrable functions with exponential decay. The decay exponent depends only on the balance between the forced term and the loss operator corresponding to the problem under consideration. Examples and applications range from elastic Boltzmann for hard spheres to inelastic Boltzmann with diffusive forcing, selfsimilar inelastic Boltzmann and inelastic shear flow. The corresponding stationary states for inelastic interactions are rigorously shown to be overpopulated with respect to classical Maxwellians. I will present the new sharp Povzner inequality and the linking to the pointwise bound for tail decay. These techniques may aid a better understanding to the regularity properties of the solutions to the underlying Boltzmann eqaution as well as the possible corrections to asymptotics for classical hydrodynamic limits.