Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum

TL;DR

This paper proves the short-term well-posedness of 3-D compressible Euler equations with a moving vacuum boundary, using a degenerate parabolic regularization approach that preserves geometric structure and avoids derivative loss.

Contribution

It introduces a novel degenerate viscosity regularization and higher-order Hardy inequality to establish existence of smooth solutions near the vacuum boundary without derivative loss.

Findings

Existence of unique solutions in Sobolev spaces up to the moving boundary.

Solutions are smooth and have no derivative loss with respect to initial data.

Method applicable to other degenerate hyperbolic systems.

Abstract

We prove well-posedness for the 3-D compressible Euler equations with moving physical vacuum boundary, with an equation of state given by the so-called gamma gas-law for gamma > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss--Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a new higher-order Hardy-type inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.