The role of density in the energy conservation for the isentropic compressible Euler equations

Abstract

In this paper, we study Onsager’s conjecture on the energy conservation for the isentropic compressible Euler equations via establishing the energy conservation criterion involving the density ϱ∈Lk(0,T;Ll(Td)). The motivation is to analyze the role of the integrability of density in energy conservation of weak solutions in this system, since almost all known corresponding results require ϱ∈L∞(0,T;L∞(Td)). Our results imply that the lower integrability of density ϱ means that more integrability of the velocity v is necessary in energy conservation. The proof relies on the Constantin–Weinan–Titi type and Lions type commutators on the mollifying kernel.