Decay estimates for large velocities in the boltzmann equation without cutoff

Abstract

We consider solutions f = f(t; x; v) to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions x 2 Td, for hard and moderately soft potentials without the angular cutoff assumption, and under the a priori assumption that the main hydrodynamic fields, namely the local mass R f dv and local energy R R fjvj2 dv and local entropy f ln f dv, are controlled along time.We establish quantitative estimates of propagation in time of "pointwise polynomial moments", i.e., supx;v f(t; x; v)(1 + jvj)q, q > 0. In the case of hard potentials, we also prove appearance of these moments for all q > 0. In the case of moderately soft potentials, we prove the appearance of low-order pointwise moments. All these conditional bounds are uniform as t goes to +1, conditionally to the bounds on the hydrodynamic fields being uniform.