Moment Inequalities and High-Energy Tails for Boltzmann Equations with Inelastic Interactions

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.

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