on the fluctuation part ${\displaystyle}{B(u,u)(t) \eqdefa u - e^{t\, \Delta}u_{0} }$ yield $$\hbox{ for any}\quad s <s' < 2s-\frac{1}{2}, \quad ({T_{*}}(u_{0})- t)^{\frac{1}{\sigma_{s'}}}\,  \| B(u,u)(t) \|_{\dot{H}^{s'}}\, \leqslant F_{s'}(M^2)$$ Moreover, for the critical case ${\displaystyle}{ = \frac{1}{2}}$, we have $$\| B(u,u)(t) \|_{\dot{B}^{\frac{1}{2}}_{2,\infty}}\, \leqslant C\,M^2.$$ ]{}*

The proof of this lemma is postpone to Section $8$. It merely stems from product laws in Besov spaces, interpolation inequalities and from judicious splitting into low and high frequencies in the following sense $$({T_{*}}-t)2^{2j} \leqslant 1 \quad  \hbox{and} \quad  ({T_{*}}-t)2^{2j} \geqslant 1.$$

Let us point out that estimates of Lemma \[fluctuation lemma\] do not hold if ${\displaystyle}{0<\alpha < \frac{1}{2}}$, owing to low frequencies. Indeed, arguments similar to the ones used in the proof of Lemma \[fluctuation lemma\] lead only to the following estimate $$\| B(u,u)(t) \|_{\dot{B}^{\alpha}_{2,\infty}}\, \leqslant C\, M^2\, {T_{*}}(u_{0})^{\frac{1}{2}(\alpha-\frac{1}{2})}.$$

0.3cm *Partial proof of Key Theorem \[Big key theorem\]*\
In all this text, we denote by $(\varepsilon_{n}) $ a non increasing sequence, which tends to $0$, when $n$ tend to $+\infty$.\
$\bullet$ Step $1$ : Existence of sup-critical elements in $ \dot{H}^s$, with ${\displaystyle}{\frac{1}{2} < s < \frac{3}{2}}\cdotp$\
Let us consider the sequence ${\displaystyle}{(M_{c} + \varepsilon_{n})_{n \geqslant 0}}$. By definition of $M_{c}$, there exists a sequence 