Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions

Abstract

This paper demonstrates the equivalence of the Euler and the Lagrangian equations of gas dynamics in one space dimension for weak solutions which are bounded and measurable in Eulerian coordinates. The precise hypotheses include all known global solutions on R × R +. In particular, solutions containing vacuum states (zero mass density) are included. Furthermore, there is a one-to-one correspondence between the convex extensions of the two systems, and the corresponding admissibility criteria are equivalent. In the presence of a vacuum, the definition of weak solution for the Lagrangian equations must be strengthened to admit test functions which are discontinuous at the vacuum. As an application, we translate a large-data existence result of DiPerna for the Euler equations for isentropic gas dynamics into a similar theorem for the Lagrangian equations.