Communications in Mathematical Sciences

Volume 11 (2013)

Number 4

Uniqueness and regularity of steady states of the Boltzmann equation for viscoelastic hard-spheres driven by a thermal bath

Pages: 851 – 906



R. J. Alonso (Department of Computational & Applied Mathematics, Rice University, Houston, Texas, U.S.A.)

B. Lods (Department of Economics and Statistics, Università degli Studi di Torino & Collegio Carlo Alberto, Torino, Italy)


We study the uniqueness and regularity of the steady states of the diffusively driven Boltzmann equation in the physically relevant case where the restitution coefficient depends on the impact velocity including, in particular, the case of viscoelastic hard-spheres. We adopt a strategy which is novel in several aspects; in particular, our study of regularity does not requires a priori knowledge of the time-dependent problem. Furthermore, the uniqueness result is obtained in the small thermalization regime by studying the so-called quasi-elastic limit for the problem. An important new aspect lies in the fact that no entropy functional inequality is needed in the limiting process.


Boltzmann equation, granular flows, viscoelastic hard-spheres, stationary solution, uniqueness, scaling, quasi-elastic limit

2010 Mathematics Subject Classification

76P05, 76T25

Published 15 June 2013