Localised Relative Energy and Finite Speed of Propagation for Compressible Flows

TL;DR

This paper extends the relative energy method to compressible Euler equations, providing new insights into weak solutions, including local uniqueness, smoothness preservation, and finite speed of propagation.

Contribution

It generalizes the relative energy principle to compressible flows and adapts classical arguments to establish local properties and finite propagation speed.

Findings

Establishment of localised relative energy principle for compressible Euler equations

Proof of local weak-strong uniqueness for solutions

Demonstration of finite speed of propagation in the isentropic system

Abstract

For the incompressible and the isentropic compressible Euler equations in arbitrary space dimension, we establish the principle of localised relative energy, thus generalising the well-known relative energy method. To this end, we adapt classical arguments of C. Dafermos to the Euler equations. We give several applications to the behaviour of weak solutions, like local weak-strong uniqueness, local preservation of smoothness, and finite speed of propagation for the isentropic system.