A regularity result for the free boundary compressible Euler equations of a liquid

TL;DR

This paper establishes a priori estimates for the free boundary compressible Euler equations modeling a liquid without surface tension, introducing a new weighted framework that improves flow map regularity without relying on higher order wave equations.

Contribution

The paper presents a novel weighted functional framework for analyzing free boundary compressible Euler equations, avoiding the need for higher order wave equations for density.

Findings

Derived a priori estimates for the equations

Introduced a new weighted functional framework

Achieved improved regularity of the flow map

Abstract

We derive a priori estimates for the compressible free boundary Euler equations in the case of a liquid without surface tension. We provide a new weighted functional framework which leads to the improved regularity of the flow map by using the Hardy inequality. One of main ideas is to decompose the initial density function. It is worth mentioning that in our analysis we do not need the higher order wave equation for the density.