The compressible Euler equations in a physical vacuum: A comprehensive Eulerian approach

Abstract

This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical interest in this system, the prior work on this problem is limited to Lagrangian coordinates, in high-regularity spaces. Instead, the objective of the present work is to provide a new, fully Eulerian approach to this problem, which provides a complete, Hadamard-style well-posedness theory for this problem in low-regularity Sobolev spaces. In particular, we give new proofs for existence, uniqueness, and continuous dependence on the data with sharp, scale-invariant energy estimates, and a continuation criterion.