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

Abstract

In present note we establish the following inequality for the the Leray-Hopf solutions of the 3-D $\Omega$-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 $\varphi :{\mathbb{R}}_{+}\to \mathbb{R}$ is an absolutely continouos non-decreasing function with bounded derivative. %with ${\varphi }^{\prime }(\xi )\ge 0$ for all $\xi >0$. Here $(\cdot ,\cdot )$ and $|\cdot |$ is correspondingly the $L^2$ inner product and the $L^2$ norm on $\Omega$, and $A$ is the Stokes operator.