# The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, III: the 3-D Boltzmann equation

## TL;DR

This paper proves the global existence and analyzes the large time behavior of smooth solutions to the 3-D Boltzmann equation in an infinitely expanding domain, showing the solution tends to vacuum without forming vacuum regions.

## Contribution

It extends previous work on compressible flows to the Boltzmann equation, establishing global existence and detailed asymptotic behavior in an expanding domain.

## Key findings

- Solutions remain smooth for all time
- Density bounds confirm tendency towards vacuum
- No vacuum formation occurs in finite time

## Abstract

This paper is a continuation of the works in \cite{Euler} and \cite{NS}, where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function.