Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential

TL;DR

This paper establishes upper bounds for solutions of the 2D homogeneous Boltzmann equation with hard potentials, demonstrating regularization and decay properties for the solution over time.

Contribution

It provides explicit upper bounds for the solution of the 2D Boltzmann equation with hard potentials, highlighting regularization effects.

Findings

Solution exhibits regularization for positive times

Upper bounds of the form f_t(v) ≤ C t^{-η} e^{-|v|^λ} are derived

Results apply to initial distributions excluding Dirac masses

Abstract

We deal with $f\_{t}(dv),$ the solution of the homogeneous $2D$ Boltzmannequation without cutoff. The initial condition $f\_{0}(dv)$ may be anyprobability distribution (except a Dirac mass). However, for sufficiently hardpotentials, 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 upperbounds for $f\_{t}(v),$ the most significant one being of type $f\_{t}(v)\leqCt^{-\eta}e^{-\left\vert v\right\vert ^{\lambda}}$ for some $\eta,\lambda>0.$