The Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions

TL;DR

This paper advances the mathematical understanding of the inhomogeneous Boltzmann equation without angular cutoff by establishing regularization, uniqueness, non-negativity, and convergence properties of solutions, completing a comprehensive theoretical framework.

Contribution

It provides the full regularization, uniqueness, and convergence results for solutions of the Boltzmann equation without angular cutoff, extending previous well-posedness results.

Findings

Solutions exhibit full regularization in all variables.

Solutions are unique and non-negative.

Convergence to equilibrium is established.

Abstract

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. Together with the results of Parts I and II about the well posedness of the Cauchy problem around Maxwellian, we conclude this series with a satisfactory mathematical theory for Boltzmann equation without angular cutoff.