Composite wave interactions and the collapse of vacuums in gas dynamics

We consider the p-system of isentropic gas dynamics. One of the outstanding questions in the study of one-dimensional Euler equations is the BV-existence and local structure of solutions having large data, including the vacuum state. The author has recently given a full description of pairwise wave interactions in 2×2 gas dynamics, which includes uniform interaction estimates up to vacuum. In this paper we consider composite interactions, which can be regarded as a degenerate superposition of pairwise interactions. We construct a class of weak solutions which demonstrate some interesting and surprising features, such as a shock of one family disappearing and a shock of the opposite family emerging. We give precise quantitative conditions which determine the outgoing waves. We also construct weak solutions of the p-system which demonstrate the collapse of a vacuum: in most cases two shocks will emerge from the vacuum, but in certain asymmetric cases a single shock and a rarefaction may emerge. We emphasize that the solutions constructed here are both explicit and exact weak solutions to the Euler equations of isentropic gas dynamics.