such solutions.\

*Hypothesis $\mathcal{H}$:* *[ There exist some blowing up $NS$-solutions such that ${\displaystyle}{({T_{*}}(u_{0}) - t)\, \| NS(u_{0}) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant M}$. ]{}*

0.1cm Notice that a very close question to this one is to prove that $$\hbox{If} \quad {T_{*}}(u_{0}) < \infty, \quad \hbox{does} \quad \limsup_{t \to {T_{*}}(u_{0}) } ({T_{*}}(u_{0}) - t)\, \| NS(u_{0})(t) \|^{\sigma_{s}}_{\dot{H}^s} = +\infty \quad ?$$

We underline that this question about blowing-up Navier-Stokes solutions has been highly developed in the context of critical spaces, namely $\dot{H}^{\frac{1}{2}}(\R^3)$ and $L^{3}(\R^3)$. Indeed, L. Escauriaza, G. Seregin and V. $\breve{\mathrm{S}}$ver$\acute{\mathrm{a}}$k showed in the fundamental work [@ESS] that any “Leray-Hopf” weak solution which remains bounded in $L^{3}(\R^3)$ can not develop a singularity in finite time. Alternatively, it means that $$\hbox{If} \,\, {T_{*}}(u_{0}) <  +\infty,\,\,\,\, \hbox{then} \,\, \limsup_{t \to {T_{*}}(u_{0})} \, \| NS(u_{0})(t) \|_{L^3} = +\infty.$$ I. Gallagher, G. Koch and F. Planchon revisited the above criteria in the context of mild Navier-Stokes solutions. They proved in [@GKP] that strong solutions which remain bounded in $L^{3}(\R^3)$, do not become singular in finite time. To perform it, they develop an alternative viewpoint : the method of “critical elements” (or “concentration-compactness”), which was introduced by C. Kenig and F. Merle to treat critical dispersive equations. 