Recently, same authors extend the method in [@GKP2] to prove the same result in the case of the critical Besov space ${\displaystyle}{\dot{B}^{-1 +\frac{3}{p}}_{p,q}(\R^3)}$, with ${\displaystyle}{3<p,q<\infty}$. Notice the work of J.-Y.Chemin and F. Planchon in [@CP], who gives the same answer in the case of the Besov space ${\displaystyle}{\dot{B}^{-1 +\frac{3}{p}}_{p,q}(\R^3)}$, with ${\displaystyle}{3<p<\infty}$, $q<3$ and with an additional regularity assumption on the data. To conclude the non-exhaustive list of blow up results, we mention the work of C. Kenig and G. Koch who carried out in [@KK] such a program of critical elements for solutions in the simpler case $\dot{H}^{\frac{1}{2}}(\R^3)$. More precisely, they proved for any data $u_{0}$ belonging to the smaller critical space  $\dot{H}^{\frac{1}{2}}(\R^3)$, $$\hbox{If} \,\, {T_{*}}(u_{0}) <  +\infty,\,\,\,\, \hbox{then} \,\, \lim_{t \to {T_{*}}(u_{0})} \, \| NS(u_{0})(t) \|_{\dot{H}^{\frac{1}{2}}} = +\infty.$$

In our case (remind : we consider Sobolev spaces $\dot{H}^{s}(\R^3)$ with ${\displaystyle}{ \frac{1}{2} < s < \frac{3}{2}}$ which are non-invariant under the natural scaling of Navier-Stokes equations), we can not expect to prove our result in the same way, because of the scaling. Indeed, a similar proof leads us to define the critical quantity $M^{\sigma_{s}}_{c}$ $$M^{\sigma_{s}}_{c} = \sup \bigl\{ A>0,\,\, \sup_{t < {T_{*}}(u_{0}) } ({T_{*}}(u_{0}) - t)\, \| NS(u_{0}) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant 