On some properties of kinetic and hydrodynamic equations for inelastic interactions

We investigate a Boltzmann equation for inelastic scattering in which the r elative velocity in the collision frequency is approximated by the thermal speed, The inelasticity is given by a velocity variable restitution coeffic ient. This equation is the analog to the Boltzmann classical equation for M axwellian molecules We study the homogeneous regime using Fourier analysis methods. We analyze the existence and uniqueness questions. the linearized operator around the Dirac delta function, self-similar solutions and moment equations. We clarify the conditions under which self-similar solutions de scribe the asymptotic behavior of the homogeneous equation. We obtain forma lly a hydrodynamic description for near elastic particles under the assumpt ion of constant and variable restitution coefficient. We describe the linea r long-wave stability/ instability for homogeneous cooling states.