Weak* Solutions II: The Vacuum in Lagrangian Gas Dynamics

TL;DR

This paper extends the weak* solution framework to include vacuum states in Lagrangian gas dynamics, allowing for rigorous treatment of vacuums represented by Dirac masses and analyzing their properties.

Contribution

It introduces a natural definition of vacuum solutions within the weak* framework and demonstrates their features through explicit examples, also extending methods to elasticity.

Findings

Vacuum states can be incorporated as Dirac masses in weak* solutions.

Explicit examples illustrate the behavior of vacuum solutions.

Extensions to elasticity show fractures cannot form in entropy solutions.

Abstract

We develop a framework in which to make sense of solutions containing the vacuum in Lagrangian gas dynamics. At and near vacuum, the specific volume becomes infinite and enclosed vacuums are represented by Dirac masses, so they cannot be treated in the usual weak sense. However, the weak* solutions recently introduced by the authors can be extended to include solutions containing vacuums. We present a definition of these natural vacuum solutions and provide explicit examples which demonstrate some of their features. Our examples are isentropic for clarity, and we briefly discuss the extension to the full $3\times3$ system of gas dynamics. We also extend our methods to one-dimensional dynamic elasticity to show that fractures cannot form in an entropy solution.