Energy identity for the incompressible Cahn--Hilliard/Navier--Stokes system with non-degenerate mobility

We consider the Cahn–Hilliard/Navier–Stokes system with non-degenerate mobility in the space-periodic case, describing the flow of two viscous immiscible and incompressible Newtonian fluids with matched densities. We identify sufficient conditions on the velocity field for weak solutions to satisfy an energy identity, improving previous results on the literature.