Regularity and long time behavior of the Boltzmann equation for granular gases

This paper investigates regularity of solutions of the Boltzmann equation with dissipative collisions in a thermal bath. In the case of pseudo-Maxwellian approximation, we prove that for any initial datum f0(ξ ) in the set of probability density with zero bulk velocity and finite temperature, the unique solution of the equation satisfies f (ξ, t) ∈ H∞(R3) for all t > 0. Furthermore, for any t0 > 0 and s 0 the Hs norm of f (ξ, t) is bounded for t t0. As a consequence, the exponential convergence to the unique steady state is also established under the same initial condition. © 2005 Elsevier Inc. All rights reserved