Decay estimates for large velocities in the Boltzmann equation without cutoff

• Published in: Journal de l'École polytechnique. Mathématiques Vol. 7; pp. 143 - 184
• Main Authors: Imbert, Cyril, Mouhot, Clément, Silvestre, Luis
• Format: Journal Article
• Language: English
• Published: École polytechnique 01.01.2020
• Subjects: Analysis of PDEs; Mathematics
• ISSN: 2429-7100; 2270-518X
• DOI: 10.5802/jep.113

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.