$\tau \in [0,1[$, $$(1-\tau)\| NS(v_{0,n})(\tau,x) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant M^{\sigma_{s}}_{c} + \varepsilon_{n}.$$ The sequence $(v_{0,n})$ satisfies the hypothesis of Lemma \[general lemma for critical element \]. Applying it, we build a sup-critical solution $ \Phi = NS(\Psi_{0}) $ in $\dot{H}^s$ which blows up at time $1$, e.g $$\limsup_{\tau \to 1} (1 - \tau)\,  \| NS(\Psi_{0})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} =\, M^{\sigma_{s}}_{c}.$$ This proves the first part of the statement of Theorem \[Big key theorem\].\
$\bullet$ Step $2$ : Existence of sup-critical elements in $\dot{H}^s \cap \dot{B}^{\frac{1}{2}}_{2,\infty} \cap \dot{H}^{s'} $, with $s$ and $s'$ such that ${\displaystyle}{s<s'< 2s -\frac{1}{2}\cdotp}$\
This will be proved in Section $6$. Notice that proving that $NS(\Psi_{0})$ is bounded in the Besov space $\dot{B}^{\frac{1}{2}}_{2,\infty}$ is equivalent to prove that $\Psi_{0}$ belongs to $\dot{B}^{\frac{1}{2}}_{2,\infty}$, since, by vertue of Lemma \[fluctuation lemma\], the fluctuation part is bounded in $\dot{B}^{\frac{1}{2}}_{2,\infty}$ and obviously we have $$\| NS(\Psi_{0})(t)\|_{\dot{B}^{\frac{1}{2}}_{2,\infty}}  \leqslant\| NS(\Psi_{0})(t) \,\,-\,\, e^{t\Delta}\Psi_{0}\|_{\dot{B}^{\frac{1}{2}}_{2,\infty}} \quad + \quad  \| e^{t\Delta}\Psi_{0}\|_{\dot{B}^{\frac{1}{2}}_{2,\infty}}.$$

The paper is structured as follows. In Section $3$, we recall the main tools of this paper. Essentially, it deals with the profile theory of P. Gérard [@PG] and a structure lemma concerning a $NS$-solution associated with a sequence which satisfies hypothesis of profile theory. We also recall some basics 