On the existence of the weak solution with local energy inequality to the 3-D inhomogeneous incompressible Navier–Stokes equations

In this paper, we show there exists a weak solution to the 3-D inhomogeneous incompressible Navier–Stokes equations satisfying in addition the local energy inequality. The same result in the homogeneous incompressible case was described in [1, Theorem 3.2].