  In this paper we give optimal lower bounds for the blow-up rate of the
$\dot{H}^{s}\left(\mathbb{T}^3\right)$-norm, $\frac{1}{2}<s<\frac{5}{2}$, of a
putative singular solution of the Navier-Stokes equations, and we also present
an elementary proof for a lower bound on blow-up rate of the Sobolev norms of
possible singular solutions to the Euler equations when $s>\frac{5}{2}$.
