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

In present note we establish the following inequality for the the Leray-Hopf solutions of the 3-D $Ω$-periodic Navier-Stokes Equations: \[ϕ(|u(t)|^2)-ϕ(|u(t_0)|^2)\le 2\int_{t_0}^{t}ϕ'(|u(τ)|^2) [-ν|A^{1/2}u(τ)|^2+(g(τ),u(τ))]\,dτ\] for all $t_0$ Leray-Hopf points, $t\ge t_0$, and $ϕ:\mathbb{R}_{+}\to\mathbb{R}$ is an absolutely continouos non-decreasing function with bounded derivative. %with $ϕ'(ξ)\ge0$ for all $ξ>0$. Here $(\cdot,\cdot)$ and $|\cdot|$ is correspondingly the $L^2$ inner product and the $L^2$ norm on $Ω$, and $A$ is the Stokes operator.