High-Velocity Tails of the Inelastic and the Multispecies Mixture Boltzmann Equations

Abstract

We study high-velocity tails of some homogeneous Boltzmann equations on v \in Rdv. First, we consider spatially homogeneous Inelastic Boltzmann equation with noncutoff collisionkernel, in the case of moderately soft potentials. We also study spatially homogeneous mixtureBoltzmann equations, for both noncutoff collision kernel with moderately soft potentials and cutoff collision kernel with hard potentials. In the case of noncutoff inelastic Boltzmann, we obtain f(t, v)≥ a(t)e b(t)| v| p , 2 < p < 6.213, by extending the cancellation lemma [R. Alexandre et al., Arch. Ration. Mech. Anal., 152 (2000), pp. 327-355] and spreading lemma [C. Imbert, C. Mouhot, and L. Silvestre, SIAM J. Math. Anal., 52 (2020), pp. 2930-2944] and assuming f \in C\infty . For the mixture-type Boltzmann equations, we prove Maxwellian p= 2.