Global existence and full regularity of the Boltzmann equation without angular cutoff

TL;DR

This paper establishes the global existence, uniqueness, and smoothness of solutions to the Boltzmann equation without angular cutoff, using a novel norm to handle the singular collision kernel.

Contribution

It introduces a new norm to effectively manage the singularity in the collision operator, proving full regularity and positivity of solutions.

Findings

Solutions are globally existent and unique.

Solutions become smooth ($C^$) for positive times.

The new norm captures the collision kernel's singularity.

Abstract

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and $C^\infty$ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.