Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

We investigate the properties of the collision operator Q associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. On deduces from this result a complete picture of the spectrum of Q in an Hilbert setting, generalizing results from Carleman to granular gases. From this integral representation of Q we also obtain that the semigroup associated to the linear Boltzmann equation for dissipative hard spheres is honest.