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Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres. (English) Zbl 1178.82056
The spatially homogeneous Boltzmann equation is considered for hard spheres undergoing inelastic collisions with a constant normal restitution coefficient in the range [0,1]. One needs to know rudiments of a physical theory of granular gases and a mathematical introduction to this theory, set by the present authors [part I with M. Rodriguez Ricard, J. Stat. Phys. 124, No. 2–4, 655–702 (2006; Zbl 1135.82325), part II with C. Mouhot, ibid., 703–746 (2006; Zbl 1135.82030)]. A brief review is given of why simplified Boltzmann models for inelastic Maxwell molecules or pseudo inealstic hard spheres do not capture some crucial physical features of the cooling process of granular gases. Basic mathematical works on spatially homogeneous inelastic hard spheres Boltzmann models are mentioned as a departure point of the analysis.
In the present paper, the self-similarity properties of the Boltzmann equation for inelastic hard spheres are studied. The uniqueness and attractivity of self-similar solutions are established, thus giving a complete answer to the Ernst-Brito conjecture [M. H. Ernst and R. Brito, J. Stat. Phys. 109, No. 3–4, 407–432 (2002; Zbl 1015.82030)] for inelastic hard spheres with small inelasticity. The proofs are based on adopting a suitable elastic limit rescaling and the construction of a smooth path of self-similar profiles. Suitable tools from perturbative theory of linear operators are employed as well.

MSC:
82C22 Interacting particle systems in time-dependent statistical mechanics
76T25 Granular flows
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C70 Transport processes in time-dependent statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35F25 Initial value problems for nonlinear first-order PDEs
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