ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

One-dimensional kinetic models of granular flows

Toscani, Giuseppe

Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy. (


We introduce and discuss a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. Then, the behavior of the Boltzmann equation in the quasi elastic limit is investigated for a wide range of the rate function. By this limit procedure we obtain a class of nonlinear equations classified as nonlinear friction equations. The analysis of the cooling process shows that the nonlinearity on the relative velocity is of paramount importance for the finite time extinction of the solution.

(Received July 11 2000)

(Revised October 11 2000)

(Online publication April 15 2002)

Key Words:

  • Granular gases;
  • Boltzmann equation;
  • long-time behavior of solutions.

Mathematics Subject Classification:

  • 76P05;
  • 82C40