Powerlike decreasing solutions of the Boltzmann equation for a Maxwell gas

We study the homogeneous, isotropic, nonlinear Boltzmann equation for a Maxwellian interaction. We show that solutions decreasing like inverse powers of the energy are physically acceptable both in the linearized and the quadratic problem. Because all moments may not exist, we introduce a generalized generating function and a finite differential system for generalized Sonine moments is derived. These new solutions may lead to small relaxation rates and justify in most cases the linear approximation.