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Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. (English) Zbl 1097.82021
Summary: We study high-energy asymptotics of the steady velocity distributions for model kinetic equations describing various regimes in dilute granular flows. The main results obtained are integral estimates of solutions of the Boltzmann equation for inelastic hard spheres, which imply that steady velocity distributions behave in a certain sense as \(C \exp(-r|v|^s)\), for \(|v|\) large. The values of \(s\), which we call the orders of tails, range from \(s = 1\) to \(s = 2\), depending on the model of external forcing. To obtain these results we establish precise estimates for exponential moments of solutions, using a sharpened version of the Povzner-type inequalities.

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
76M35 Stochastic analysis applied to problems in fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76T25 Granular flows
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