Localised relative energy and finite speed of propagation for compressible flows

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.