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.