The dissipative linear Boltzmann equation for hard spheres

Abstract

The authors prove existence and uniqueness of a Maxwellian, normalized equilibrium state for a dissipative linear Boltzmann equation with hard-sphere collision kernel modeling a granular gas and, for initial data with finite temperature and entropy, strong L 1 convergence (obtained through compactness arguments) toward the equilibrium of the solutions in the space-homogeneous case. The form of the equilibrium state, which is universal for the family of collision operators including hard, soft and Maxwellian interactions, is guessed through a grazing collision asymptotics and then proved to be the equilibrium state through Fourier analysis. Uniqueness and strong L 1 convergence proofs follow the procedure in [R. Pettersson, J. Statist. Phys. 72 (1993), no. 1-2, 355–380; MR1233035 (94e:82095)] and in a cited preprint by the same author ["On solutions to the linear Boltzmann equation for granular gases'', Preprint No. 2003:41, Dept. Math., Chalmers Univ., Göteborg, Sweden, 2003, available at www.math.chalmers.se/Math/Research/Preprints/]. Reviewed by Valeria Ricci