Convergence to self-similar solutions for the homogeneous Boltzmann equation

Abstract

The Boltzmann H-theorem implies that the solution to the Boltzmann equation tends to an equilibrium, that is, a Maxwellian when time tends to infinity. This has been proved in varies settings when the initial energy is finite. However, when the initial energy is infinite, the time asymptotic state is no longer described by a Maxwellian, but a self-similar solution obtained by Bobylev–Cercignani. The purpose of this paper is to rigorously justify this for the spatially homogeneous problem with a Maxwellian molecule type cross section without angular cutoff.