Well-posedness for compressible Euler equations with physical vacuum singularity

TL;DR

This paper establishes local well-posedness for 1D compressible Euler equations with physical vacuum singularities, using new formulations and energy spaces to handle degeneracy at the vacuum boundary.

Contribution

It introduces a novel formulation and energy spaces to prove well-posedness for Euler equations with physical vacuum singularities.

Findings

Proves local in time well-posedness for the problem.

Develops new energy spaces suited to the vacuum singularity.

Handles degeneracy at the vacuum boundary effectively.

Abstract

An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristics coincide and have unbounded derivative. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local in time well-posedness of one-dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity.