  Considering initial data in $\dot{H}^s$, with $\frac{1}{2} \textless{} s
\textless{} \frac{3}{2}$, this paper is devoted to the study of possible
blowing-up Navier-Stokes solutions such that $(T*(u\_{0}) -t)^{\frac{1}{2} (s-
\frac{1}{2})} \,\, \| u \|\_{\dot{H}^s}}$ is bounded. Our result is in the
spirit of the tremendous works of L. Escauriaza, G. Seregin, and V.
$\breve{\mathrm{S}}$ver$\acute{\mathrm{a}}$k and I. Gallagher, G. Koch, F.
Planchon, where they proved there is no blowing-up solution which remain
bounded in $L^3(R^3)$. The main idea is that if such blowing-up solutions
exist, they satisfy critical properties.
