Energy conservation for the weak solutions of the compressible Navier-Stokes equations

In this talk, we discuss the energy conservation for the weak solutions of the compressible Navier-Stokes equations globally in time, under certain conditions. The results hold for the renormalized solutions of the equations with constant viscosities, as well as the weak solutions of the equations with degenerate viscosity. The energy may conserve on the vacuum for the compressible Navier-Stokes equations with constant viscosities. Our results are the first ones on the energy conservation for the weak solutions of the compressible Navier-Stokes equations.