SUFFICIENT CONDITIONS FOR LOCAL ENERGY CONSERVATION FOR THE COMPRESSIBLE EULER SYSTEM

We prove a local energy equality for the compressible isentropic Euler equations under the assumptions of Besov regularity in space and Bochner integrability in time. We show that this result can be established for a certain range of exponents which control regularity. We use the Littlewood-Paley decomposition to extend the energy equality to a borderline value of the regularity exponent.