A general sufficient criterion for energy conservation in the Navier-Stokes system

In this paper, we derive an energy conservation criterion based on a combination of velocity and its gradient for the weak solutions of both the homogeneous incompressible Navier-Stokes equations and the general compressible Navier-Stokes equations. For the incompressible case, this class implies most known corresponding results on periodic domain via either the velocity or its gradient including the famous Lions' energy conservation criterion obtained in \cite{[Lions]}. For the compressible case, this helps us to extend the previously known criteria for the energy conservation of weak solutions from the incompressible fluid to compressible flow and improve the recent results due to Nguyen-Nguyen-Tang in \cite[Nonlinearity 32 (2019)]{[NNT]} and Liang in \cite[Proc. Roy. Soc. Edinburgh Sect. A (2020)]{[Liang]}.