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

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

2006-07-21

arxiv.org/abs/math/0607536v1

Journal of Statistical Physics 124 (2006) 703-746

Mathematics

Analysis of PDEs

41 pages

Scientific paper

10.1007/s10955-006-9097-8

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.

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