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∈𝕋 d , 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 ∫fdv and local energy ∫f|v| 2 dv and local entropy ∫flnfdv, are controlled along time. We establish quantitative estimates of propagation in time of “pointwise polynomial moments”, i.e., sup x,v f(t,x,v)(1+|v|) 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 +∞, conditionally to the bounds on the hydrodynamic fields being uniform.