  We consider the incompressible inhomogeneous Navier-Stokes equations with
constant viscosity coefficient and density which is bounded and bounded away
from zero. We show that the energy balance relation for this system holds for
weak solutions if the velocity, density, and pressure belong to a range Besov
spaces of smoothness $1/3$. A density-dependent version of the classical
K\'arm\'an-Howarth-Monin relation is derived.
