On a class of self-similar solutions of the Boltzmann equation

Abstract

We consider a class of distribution functions having the form $ f(v,t)=t^{-dat} F(v e^{-at} )$, $ a=\mathrm{const.} $, where $ v $ and $ t $ denote the particle velocity and the time. This class of self-similar solutions to the spatially homogeneous Boltzmann equation (BE) for Maxwell molecules was studied by Bobylev and Cercignani in early 2000s. The solutions are positive, but have an infinite second moment (energy). However, the same class of distribution functions with finite energy appears to be closely connected with quite different class of group-invariant solutions of the spatially inhomogeneous BE. This is a motivation for considering the so-called modified spatially homogeneous BE, which contains an extra force term proportional to a matrix $ A $. The modified BE was recently studied under assumption of "sufficient smallness of norm $ \|A\| $" without explicit estimates of the smallness. We fill this gap and prove that all important facts related to self-similar solutions remain valid also for moderate values of $ \|A\| $.