Expansion of a compressible gas in vacuum

TL;DR

This paper investigates the behavior of gas fronts in vacuum for the isentropic Euler system, identifying conditions for acceleration or constant velocity, and contrasting solutions in even and odd space dimensions.

Contribution

It provides a sufficient condition for the emergence of boundary singularities and demonstrates the non-existence of eternal non-accelerated flows in odd dimensions.

Findings

Eternal non-accelerated flows exist in even dimensions under certain initial conditions.

Boundary singularities appear when initial data meet specific criteria.

In one dimension, asymptotic behaviors at infinite times are closely related.

Abstract

Tai-Ping Liu \cite{Liu\_JJ} introduced the notion of "physical solution' of the isentropic Euler system when the gas is surrounded by vacuum. This notion can be interpreted by saying that the front is driven by a force resulting from a H\"older singularity of the sound speed. We address the question of when this acceleration appears or when the front just move at constant velocity. We know from \cite{Gra,SerAIF} that smooth isentropic flows with a non-accelerated front exist globally in time, for suitable initial data. In even space dimension, these solutions may persist for all $t\in\R$ ; we say that they are {\em eternal}. We derive a sufficient condition in terms of the initial data, under which the boundary singularity must appear. As a consequence, we show that, in contrast to the even-dimensional case, eternal flows with a non-accelerated front don't exist in odd space dimension. In one space dimension, we give a refined definition of physical solutions. We show that for a shock-free flow, their asymptotics as both ends $t\rightarrow\pm\infty$ are intimately related to each other.