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

TL;DR

This paper investigates self-similar solutions of the Boltzmann equation, extending previous results to cases with moderate force terms, and explores their connection to inhomogeneous solutions.

Contribution

It proves that key properties of self-similar solutions hold for the modified Boltzmann equation with moderate force matrix norms.

Findings

Self-similar solutions have infinite energy in the classical case.

Modified Boltzmann equation with force term admits similar solutions for moderate force strength.

Connections between homogeneous and inhomogeneous solutions are clarified.

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\| $.