Lower bounds for possible singular solutions for the Navier--Stokes and Euler equations revisited

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 $s>\frac{5}{2}$.