statements will take place in some Sobolev and Besov spaces, with a regularity index $s$ such that ${\displaystyle}{\frac{1}{2} < s < \frac{3}{2}\cdotp}$\
**Notations.** We shall constantly be using the following simplified notations: $$L^{\infty}_{T}(\dot{H}^s)  \eqdefa L^{\infty}([0,T],\dot{H}^s)\quad \hbox{and} \quad L^{2}_{T}(\dot{H}^{s+1})  \eqdefa L^{2}([0,T],\dot{H}^{s+1}),$$ and the relevant function space we shall be working with in the sequel is $$X^{s}_{T} \eqdefa L^{\infty}_{T}(\dot{H}^s)\, \cap\, L^{2}_{T}(\dot{H}^{s+1}),\quad \hbox{endowed with the norm} \quad\| u \|^2_{X^{s}_{T}}  \eqdefa \| u \|^2_{L^{\infty}_{T}(\dot{H}^s)} + \| u \|^2_{L^{2}_{T}(\dot{H}^{s+1})}.$$ 0.2cm Let us start by recalling the local existence theorem for data in the Sobolev space $\dot{H}^s$.

*[ \[key theorem\] Let $u_{0}$ be in $\dot{H}^s$, with ${\displaystyle}{\frac{1}{2} < s < \frac{3}{2}}\cdotp$ Then there exists a time $T$ and there exists a unique solution $NS(u_{0})$ such that ${\displaystyle}{NS(u_{0}) \quad\hbox{belongs to} \quad L^{\infty}_{T}(\dot{H}^s) \cap L^{2}_{T}(\dot{H}^{s+1})}$.\
Moreover, denoting by ${T_{*}}(u_{0})$ the maximal time of existence of such a solution, there exists a positive constant  $c$ such that $$\label{relation Ac et ts}
 {T_{*}}(u_{0})\,\, \| u_{0} \|^{\sigma_{s}}_{\dot{H}^s} \geqslant c,\quad \hbox{with} \quad \sigma_{s} \eqdefa \frac{1}{\frac{1}{2}(s-\frac{1}{2})}\cdotp$$]{}*

Throughout this paper, we will adopt the useful notation $NS(u_{0})$ to mean the maximal solution of the Navier-Stokes system, associated with the initial data $u_{0}$. Notice that our whole work relies on the hypothesis there exists some 