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

Stéphane Mischler (1) , Clément Mouhot (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

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

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Stéphane Mischler, Clément Mouhot. Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior. Journal of Statistical Physics, 2006, 124 (2-4), pp.703-746. ⟨10.1007/s10955-006-9097-8⟩. ⟨hal-00087235⟩