Decay estimates for large velocities in the Boltzmann equation without cutoff
Résumé
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.
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