Energy Equality for the Isentropic Compressible Navier-Stokes Equations Without Upper Bound of the Density

Abstract

In this paper, we are concerned with the minimal regularity of both the density and the velocity for the weak solutions keeping energy equality in the isentropic compressible Navier-Stokes equations. The energy equality criteria without upper bound of the density are established for the first time. Our results imply that the lower integrability of the density $\rho$ means that more integrability of the velocity $v$ or the gradient of the velocity $∇v$ are necessary for energy conservation of the isentropic compressible fluid and the inverse is also true.