Let $A$ be the Stokes operator.
We show as the main result of the paper that if $w$ is a global weak solution to the Navier-Stokes equations
satisfying the strong energy inequality, $\beta \in [0,1/2]$ and $\alpha \in [\beta,\infty)$, then there exist $t_0 \ge 1$, $C_1>1$ and $\delta_1
\in (0,1)$ such that