The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, I: 3D Euler equations

TL;DR

This paper proves the global existence and describes the large-time behavior of smooth compressible inviscid fluids in an infinitely expanding ball, showing the density tends to vacuum without forming vacuum domains.

Contribution

It establishes the first rigorous confirmation of the physical phenomenon for inviscid fluids in an expanding domain, providing bounds on the density function.

Findings

Density tends to vacuum as time approaches infinity.

No vacuum domains form within the expanding ball.

Provides bounds on the density function over time.

Abstract

We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and 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 this paper, as the first part of our three papers, we will confirm this physical phenomenon for the compressible inviscid fluids by obtaining the exact lower and upper bound on the density function.