Cooling process for inelastic Boltzmann equations  for hard spheres, Part II: Self-similar solutions and  tail behavior

Mouhot, Clément; Mischler, Stéphane

Mouhot, Clément; Mischler, Stéphane (2006), Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior, Journal of Statistical Physics, 124, 2-4, p. 703-746. http://dx.doi.org/10.1007/s10955-006-9097-8

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00087235/en/

Date

2006

Journal name

Journal of Statistical Physics

Volume

124

Number

2-4

Pages

703-746

Abstract (EN)

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients. We prove the existence of self-similar solutions, and we give pointwise estimates on their tail. We also give general estimates on the tail and the regularity of generic solutions. In particular we prove Haff 's law on the rate of decay of temperature, as well as the algebraic decay of singularities. The proofs are based on the regularity study of a rescaled problem, with the help of the regularity properties of the gain part of the Boltzmann collision integral, well-known in the elastic case, and which are extended here in the context of granular gases.

Subjects / Keywords

tail behavior; regularity of the collision operator; self-similar solutions; Haff's law; cooling process; granular gas; inelastic hard spheres; Boltzmann equation