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 $x \in \mathbb T^d$, for hard and moderately soft potentials \emph{without the angular cutoff assumption}, and under the \emph{a priori} assumption that the main hydrodynamic fields, namely the local mass $\int\_v f(t,x,v)$ and local energy $\int\_v f(t,x,v)|v|^2$ and local entropy $\int\_v f(t,x,v) \ln f(t,x,v)$, are controlled along time. We establish quantitative estimates of \emph{propagation} in time of "pointwise polynomial moments", i.e. $\sup\_{x,v} f(t,x,v) (1+|v|)^q$, $q \ge 0$. In the case of hard potentials, we also prove \emph{appearance} of these moments for all $q \ge 0$. In the case of moderately soft potentials we prove the \emph{appearance} of low-order pointwise moments.