We recall a crucial property of the Navier-Stokes equation : the scaling invariance. Let us define the operator $$\label{notation}
\begin{split}
\forall \alpha \in \R^{+},\,\, \forall \lambda \in \R^{+}_{*},\,\,  &\forall x_{0} \in \R^3,\,\,\,\, \Lambda^{\alpha}_{\lambda,x_{0}}\,u(t,x) \eqdefa \frac{1}{\lambda^{\alpha}}u\Bigl(\frac{t}{\lambda^2} \virgp \frac{x-x_{0}}{\lambda}  \Bigr).\\
&\hbox{If} \,\,\, \alpha =1,\,\,\, \hbox{we note} \,\,\,  \Lambda^{1}_{\lambda,x_{0}} =  \Lambda_{\lambda,x_{0}}. 
\end{split}$$

Clearly, if $u$ is smooth solution of Navier-Stokes system on  $[0,T] \times \R^3$ with pressure $p$ associated with the initial data $u_{0}$, then, for any positive $\lambda$, the vector field and the pressure $$u_{\lambda}  \eqdefa \Lambda_{\lambda,x_{0}}\,u \quad \hbox{and} \quad p_{\lambda}  \eqdefa \Lambda^{2}_{\lambda,x_{0}}\,p$$ is a solution of Navier-Stokes system on the interval $[0,\lambda^2T] \times \R^3$, associated with the initial data $$u_{0,\lambda}  = ~\Lambda_{\lambda,x_{0}}\, u_{0}.$$ This leads to the definition of scaling invariant space.

*[ A Banach space $X$ is said to be scaling invariant (or also critical), if its norm is invariant under the scaling transformation defined by $u \mapsto u_{\lambda}$ $$|| u_{\lambda} ||_{X} = || u ||_{X}.$$]{}*

Let us give some exemples of critical spaces in dimension $3$ $$\dot{H}^{\frac{1}{2}}(\R^3) \hookrightarrow L^3(\R^3) \hookrightarrow \dot{B}^{-1 +\frac{3}{p}}_{p,\infty}(\R^3)_{p < \infty} \hookrightarrow \mathcal{BMO}^{-1}(\R^3) \hookrightarrow \dot{B}^{-1 }_{\infty,\infty}(\R^3).$$

The framework of this work is functional spaces which are above the natural scaling of Navier-Stokes equations. More precisely, our 