Expansion of a Compressible Gas in Vacuum
by Denis Serre  

Vol. 10 No. 4 (2015) P.695~P.716

ABSTRACT

Tai-Ping Liu [12] 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$\ddot{o}$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 [7, 17] that smooth isentropic ows with a non-accelerated front exist globally in time, for suitable initial data. In even space dimension, these solutions may persist for all $t\,\epsilon \,\mathbb{R}$ ; we say that they are 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 ows with a non-accelerated front do not exist in odd space dimension.

In one space dimension, we give a refined definition of physical solutions. We show that for a shock-free ow, their asymptotics as both ends $t\rightarrow \pm \infty$ are intimately related to each other.

KEYWORDS
Euler equations, global solutions, vacuum, physical singularity.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 76N15, 35L65, 35Q31.

MILESTONES

Received: 2015-03-17
Revised : 2015-06-29
Accepted: 2015-06-29

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