Hydrodynamic limits of Vlasov-Fokker-Planck equation for granular media

Abstract

This paper, which is a sequel to Benedetto-Caglioti-Golse-Pulvirenti, Comput. Math. Appl. 38 (1999), p. 121-131, considers as a starting point a mean-field equation for the dynamics of a gas of particles interacting via dissipative binary collisions. More precisely, we are concerned with the case where these particles are immersed in a thermal bath modeled by a linear Fokker-Planck operator. Two different scalings are considered for the resulting equation. One concerns the case of a thermal bath at finite temperature and leads formally to a nonlinear diffusion equation. The other concerns the case of a thermal bath at infinite temperature and leads formally to an isentropic Navier-Stokes system. Both formal limits rest on the mathematical properties of the linearized mean-field operator which are established rigorously, and on a Hilbert or Chapman-Enskog expansion.