Inelastic Boltzmann equation driven by a particle thermal bath

Abstract

We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient $ \alpha\in(0,1) $, under the thermalization induced by a host medium with fixed $ e\in(0,1] $ and a fixed Maxwellian distribution. When the restitution coefficient $ \alpha $ is close to 1 we prove existence and uniqueness of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.