Kinetic and hydrodynamic models of nearly elastic granular flows

We introduce and discuss certain models of dilute granular systems of spheres with dissipative collisions and variable coefficient of restitution, under the assumption of weak inelasticity. The dissipation is taken into account by introducing a correction to the Boltzmann collision operator in the form of a nonlinear friction type operator. Using this correction we obtain formally from the Boltzmann equation in a direct way a hydrodynamic description of a system of nearly elastic particles colliding with a variable coefficient of restitution. In one dimension of the velocity variable, the correction reduces to the nonlinear friction operator obtained in [24] as the quasi-elastic limit of a model Boltzmann equation for partially anelastic spheres. The large–time asymptotic of this one–dimensional model can be described in detail.