A priori estimates for the free-boundary 3-D compressible Euler equations in physical vacuum

TL;DR

This paper establishes a priori estimates for the 3D compressible Euler equations with a physical vacuum boundary, addressing the challenges posed by degeneracy and boundary behavior.

Contribution

It provides the first a priori estimates for the free-boundary compressible Euler equations with physical vacuum conditions in three dimensions.

Findings

Derived estimates ensure well-posedness under physical vacuum conditions.

Addressed degeneracy at the vacuum boundary in a hyperbolic free-boundary system.

Extended mathematical techniques to handle characteristic hyperbolic systems with vacuum boundaries.

Abstract

We prove a priori estimates for the three-dimensional compressible Euler equations with moving {\it physical} vacuum boundary, with an equation of state given by $p(\rho) = C_\gamma \rho^\gamma $ for $\gamma >1$. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic {\it free-boundary} system to which standard methods of symmetrizable hyperbolic equations cannot be applied.