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 $\int \_vf(t,x,v)$ and local energy $\int \_vf(t,x,v)|v{|}^2$ and local entropy $\int \_vf(t,x,v)\ln f(t,x,v)$, are controlled along time. We establish quantitative estimates of \emph{propagation} in time of "pointwise polynomial moments", i.e. $\sup \_x,vf(t,x,v)(1+|v|{)}^q$, $q\ge 0$. 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.

32 pages

References in corpus (1)

• Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation

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• Quantitative pointwise estimates of the cooling process for inelastic Boltzmann equation