  It is well-known that a Leray-Hopf weak solution in $L^4 (0,T; L^4(\Omega))$
for the incompressible Navier-Stokes system is persistence of energy due to
Lions [19]. In this paper, it is shown that Lions's condition for energy
balance is also valid for the weak solutions of the isentropic compressible
Navier-Stokes equations allowing vacuum under suitable integrability conditions
on the density and its derivative. This allows us to establish various
sufficient conditions implying energy equality for the compressible flow as
well as the non-homogenous incompressible Navier-Stokes equations. This is an
improvement of corresponding results obtained by Yu in [32, Arch. Ration. Mech.
Anal., 225 (2017)], and our criterion via the gradient of the velocity
partially answers a question posed by Liang in [18, Proc. Roy. Soc. Edinburgh
Sect. A (2020)].

Analysis of PDEs