An Eulerian-Lagrangian Formulation of the Compressible Euler Equations with Vacuum

TL;DR

This paper introduces a new Eulerian-Lagrangian approach for the compressible Euler equations with vacuum, enabling short-time solutions without special symmetrization, addressing a longstanding open problem.

Contribution

It develops a novel formulation that handles vacuum in compressible Euler equations, providing short-time solutions for general pressure laws without symmetrization.

Findings

Well-defined in vacuum conditions with compactly supported initial data

Achieves short-time solutions for general pressure laws

Addresses an open problem in compressible gas dynamics

Abstract

In this paper, we present a novel Eulerian-Lagrangian formulation for the compressible isentropic Euler equations with vaccum. Using the developed Lagrangian flow map formulation, we show a short-time solution for a general pressure law. A particularly appealing feature of the approach used, it is well defined in the presence of vacuum, namely for compactly supported initial data which constitute an important problem in gas dynamics. Moreover, it does so without relying on any special symmetrization. While analogous results are well understood for incompressible fluids, the compressible setting, particularly in the presence of vacuum, remained open.