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Cooling process for inelastic Boltzmann equations for hard spheres. II: Self-similar solutions and tail behavior. (English) Zbl 1135.82030
Summary: This is the second part of our work with M. Rodriguez Ricard [cf. Part I, J. Stat. Phys. 124, No. 2-4, 655–702 (2006; Zbl 1135.82325)].
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.

82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D05 Statistical mechanics of gases
82B40 Kinetic theory of gases in equilibrium statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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