A particle model that conserves the measure in the phase space, but does not conserve the kinetic energy

We consider a particular model of hard spheres that collide inelastically, losing a fixed amount of kinetic energy at each collision. We show that the transport associated to this hard sphere dynamics preserves locally the measure in the phase space. We prove the analog of Alexander's theorem for our model, providing the global well-posedness of the trajectories, for almost every initial datum.