Weak* Solutions II: The Vacuum in Lagrangian Gas Dynamics

We develop a framework in which to make sense of solutions containing the vacuum in Lagrangian gas dynamics. At and near a 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.