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. One deduces from this result a complete picture of the spectrum of Q in an Hilbert space setting, generalizing results from T. Carleman [5] to granular gases. In the same way, we obtain from this integral representation of Q that the semigroup in L(R ×R, dx ⊗ dv) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from [1].