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In present note we establish the following inequality for the the Leray-Hopfsolutions of the 3-D $\Omega$-periodic Navier-Stokes Equations:\[\phi(|u(t)|^2)-\phi(|u(t_0)|^2)\le 2\int_{t_0}^{t}\phi'(|u(\tau)|^2)[-\nu|A^{1/2}u(\tau)|^2+(g(\tau),u(\tau))]\,d\tau\] for all $t_0$ Leray-Hopfpoints, $t\ge t_0$, and $\phi:\mathbb{R}_{+}\to\mathbb{R}$ is an absolutelycontinouos non-decreasing function with bounded derivative. %with$\phi'(\xi)\ge0$ for all $\xi>0$. Here $(\cdot,\cdot)$ and $|\cdot|$ iscorrespondingly the $L^2$ inner product and the $L^2$ norm on $\Omega$, and $A$is the Stokes operator.

A strengthening of the energy inequality for the Leray-Hopf solutions of the 3D periodic Navier-Stokes equations