# Convergence to the Self-similar Solutions to the Homogeneous Boltzmann Equation

## TL;DR

This paper rigorously proves that solutions to the spatially homogeneous Boltzmann equation with infinite initial energy converge to self-similar solutions, extending understanding beyond finite energy cases.

## Contribution

It provides a rigorous mathematical justification for convergence to self-similar solutions in the infinite energy setting for the Boltzmann equation without angular cutoff.

## Key findings

- Solutions tend to self-similar profiles over time
- Extension of convergence results to infinite energy initial data
- Analysis specific to Maxwellian molecule type cross sections

## 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 Maxwellian molecule type cross section without angular cutoff.