Free Cooling and High-Energy Tails of Granular Gases with Variable Restitution Coefficient

We develop in this paper the first systematic treatment of the homogeneous Boltzmann equation for granular gases with non constant restitution coefficient, generalizing a large number of the results obtained recently for homogeneous granular gases with constant restitution coefficient to a broader class of physical restitution coefficients that depend on the collision impact velocity. Our analysis is carried along the following paths: first, we develop the L-theory which is based on the understanding of the moments of solution and leads as byproduct to the Haff’s law and the so-called L-exponential tails theorem. Second, we investigate the L-theory for 1 < p < ∞, proving in particular the propagation of L norms. Finally, we develop the L-theory which produces the celebrated L-exponential tails theorem as ultimate goal. In all the above steps, the study of the self-similar solutions to the Boltzmann equation plays a crucial role.