  Under assumption that $T^{\ast}$ is the maximal time of existence of smooth
solution of the 3D Navier-Stokes equations in the Sobolev space $H^{s}$, we
establish lower bounds for the blow-up rate of the type$\ \left( T^{\ast
}-t\right) ^{-\varphi\left( n\right) }$, where $n$ is a natural number
independent of $s$ and $\varphi$ is a linear function. Using this new type in
the 3D Navier-Stokes equations in the $H^{5/2}$, both on the whole space and in
the periodic case, we give an answer to a question left open by James et al
(2012, J. Math. Phys.). We also prove optimal lower bounds for the blow-up rate
in $\dot{H}^{3/2}$ and in $H^{1}$.
