Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian

TL;DR

This paper proves the existence of global smooth solutions to the non-cutoff Boltzmann equation near equilibrium by combining convergence, regularity, and short-time existence results, especially for initial data close to a Maxwellian.

Contribution

It provides a simplified proof of global existence for solutions near Maxwellian, integrating known results with a new short-time existence for weighted initial data.

Findings

Global smooth solutions exist near Maxwellian equilibrium.

Short-time existence established for polynomially-weighted initial data.

Solutions remain smooth for all time if initial data is sufficiently close to equilibrium.

Abstract

The purpose of this paper is to show how the combination of the well-known results for convergence to equilibrium and conditional regularity, in addition to a short-time existence result, lead to a quick proof of the existence of global smooth solutions for the non cutoff Boltzmann equation when the initial data is close to equilibrium. We include a short-time existence result for polynomially-weighted $L^\infty$ initial data. From this, we deduce that if the initial data is sufficiently close to a Maxwellian in this norm, then a smooth solution exists globally in time.