Physical Vacuum Problems for the Full Compressible Euler Equations: Low-regularity Hadamard-style Local Well-posedness

TL;DR

This paper proves local well-posedness for the full compressible Euler equations with physical vacuum in low-regularity spaces, allowing unbounded interface curvature, and provides energy estimates and criteria for solution continuation.

Contribution

It introduces a novel Eulerian framework for analyzing low-regularity physical vacuum problems, avoiding traditional Lagrangian regularity issues.

Findings

Established existence and uniqueness of solutions.

Derived sharp a priori energy estimates.

Provided continuation criteria for solutions.

Abstract

This manuscript concerns the dynamics of non-isentropic compressible Euler equations in a physical vacuum. We establish the Hadamard-style local well-posedness in low-regularity weighted Sobolev spaces, where the gas-vacuum interface is allowed to have unbounded curvature, demonstrating existence, uniqueness, and continuous dependence on initial data. Additionally, we prove sharp a priori energy estimates and continuation criteria. The approach is based on the framework of Eulerian coordinates, avoiding the regularity issues of the flow map and the high nonlinearity induced by the Lagrangian transformation.