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This paper addresses the problem of energy conservation for the two- andthree-dimensional density-dependent Euler equations. Two types of sufficientconditions on the regularity of solutions are provided to ensure theconservation of total kinetic energy on the entire time interval including theinitial time. The first class of data assumes integrability on the spatialgradient of the density, and hence covers the classical result ofConstantin-E-Titi for the homogeneous Euler equations. The other type of dataimposes extra time Besov regularity on the velocity profile, and thecorresponding result can be applied to deal with a wide class of rough densityprofiles.

Onsager's energy conservation for inhomogeneous Euler equations