blowing up $NS$-solutions, e.g some $NS$-solutions with a finite lifespan ${T_{*}}(u_{0})$. This is still an open question.

We point out that the infimum of the quantity ${\displaystyle}{{T_{*}}(u_{0})\,\, \| u_{0} \|^{\sigma_{s}}_{\dot{H}^s}}$ exists and is positive (because of the constant $c$). It has been proved in [@P] that there exists some intial data which reach this infimum and that the set of such data is compact, up to dilations and translations.

Theorem \[key theorem\] implies there exists a constant $c>0$, such that $$\label{remark c}
({T_{*}}(u_{0}) - t)\, \| NS(u_{0})(t) \|^{\sigma_{s}}_{\dot{H}^s} \geqslant c,$$ and thus we get in particular the blow up of the $\dot{H}^s$-norm $$\lim_{t \to {T_{*}}(u_{0})} \| NS(u_{0})(t)\|^{\sigma_{s}}_{\dot{H}^s} = +\infty.$$

0.2cm Our motivation here is to wonder if there exist some Navier-Stokes solutions which stop living in finite time (e.g ${T_{*}}(u_{0}) < \infty$) and which blows up at a minimal rate, namely: there exists a positive constant $M$ such that ${\displaystyle}{({T_{*}}(u_{0}) - t)\, \| NS(u_{0}) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant M}$. In others terms,\

*Question:* *[ Does there exist some blowing up $NS$-solutions such that ${\displaystyle}{({T_{*}}(u_{0}) - t)\, \| NS(u_{0}) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant M}$ ?\
If yes, what do they look like ? ]{}*

0.3cm We assume an affirmative answer and we search to characterize 