Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential
Abstract
We deal with $f_{t}(dv),$ the solution of the homogeneous $2D$ Boltzmann
equation without cutoff. The initial condition $f_{0}(dv)$ may be any
probability distribution (except a Dirac mass). However, for sufficiently hard
potentials, the semigroup has a regularization property (see \cite{[BF]}):
$f_{t}(dv)=f_{t}(v)dv$ for every $t>0.$ The aim of this paper is to give upper
bounds for $f_{t}(v),$ the most significant one being of type $f_{t}(v)\leq
Ct^{-\eta}e^{-\left\vert v\right\vert ^{\lambda}}$ for some $\eta,\lambda>0.$
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