critical element \] Let $(v_{0,n})_{n \in \N}$ be a bounded sequence in $\dot{H}^s$ such that $$\tau^{*}(v_{0,n}) = 1 \quad \hbox{ and} \quad \hbox{ for any} \,\,\, \tau < 1, \quad (1-\tau)\, \| NS(v_{0,n})(\tau,\cdotp) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant\,\, M^{\sigma_{s}}_{c} + \varepsilon_{n},$$ where $\varepsilon_{n}$ is a generic sequence which tends to $0$ when $n$ goes to $+\infty$.\
Then, there exists $\Psi_{0}$ in $\dot{H}^s$ such that $\Psi \eqdefa NS(\Psi_{0})$ is a sup-critical solution blowing up at time $1$ and satisfies $$\sup_{\tau < 1} (1 - \tau)\,  \| NS(\Psi_{0})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} \,=\,    \limsup_{\tau \to 1} (1 - \tau)\,  \| NS(\Psi_{0})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} =\, M^{\sigma_{s}}_{c}.$$ Moreover, the initial data of such element is a weak limit of the sequence $(v_{0,n})$ translated, e.g $${\displaystyle}{\exists\,\, (x_{0,n})_{n \geqslant 0}, \quad   v_{0,n}(\cdotp + x_{0,n} ) \rightharpoonup_{n \to +\infty} \Psi_{0}}.$$ ]{}*

The proof of Lemma \[general lemma for critical element \] will be the purpose of Section $4$. It relies essentially on scaling argument and profile theory, which will be introduced in the next Section $3$.

*[(Fluctuation estimates)\
\[fluctuation lemma\] Let $u=NS(u_{0})$ be a NS-solution associated with a data $ u_{0} \in  \dot{H}^s $, with ${\displaystyle}{\frac{1}{2} < s < \frac{3}{2}}$, such that $$({T_{*}}(u_{0}) - t)^{\frac{1}{\sigma_{s}}}\,  \| NS(u_{0})(t) \|_{\dot{H}^s} \leqslant M.$$ Then, the following estimates 