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

TL;DR

This paper establishes a unified energy conservation criterion for weak solutions of both incompressible and compressible Navier-Stokes equations, extending and improving upon previous results in the field.

Contribution

It introduces a general criterion based on velocity and its gradient that encompasses known conditions and extends them to compressible flows.

Findings

Unifies energy conservation criteria for incompressible and compressible Navier-Stokes equations.

Extends previous incompressible results to the compressible case.

Improves recent criteria by Nguyen-Nguyen-Tang and Liang.

Abstract

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]}.