in our approach by numerous works dedicated to the incompressible Navier-Stokes equation (see e.g [@Meyer]): $$\begin{cases}
\begin{aligned}
&{\partial}_{t}v+v\cdot{\nabla}v-\mu{\Delta}v+{\nabla}\Pi=0,\\
&{\rm div}v=0,
\end{aligned}
\end{cases}
\leqno{(NS)}$$ we aim at solving (\[0.1\]) in the case where the data $(\rho_{0},u_{0},f)$ have *critical* regularity.\
By critical, we mean that we want to solve the system functional spaces with norm in invariant by the changes of scales which leaves (\[0.1\]) invariant. In the case of barotropic fluids, it is easy to see that the transformations: $$(\rho(t,x),u(t,x))\longrightarrow (\rho(l^{2}t,lx),lu(l^{2}t,lx)),\;\;\;l\in{\mathbb{R}},
\label{1}$$ have that property, provided that the pressure term has been changed accordingly.

Let $\bar{\rho}>0$. In the sequel we will note: $q=\frac{\rho-\bar{\rho}}{\bar{\rho}}$.

The use of critical functional frameworks led to several new weel-posedness results for compressible fluids (see [@DL; @DW; @H1; @H2]). In addition to have a norm invariant by (\[1\]), appropriate functional space for solving (\[0.1\]) must provide a control on the $L^{\infty}$ norm of the density (in order to avoid vacuum and loss of ellipticity). For that reason, we restricted our study to the case where the initial data $(\rho_{0},u_{0})$ and external force $f$ are such that, for some positive constant $\bar{\rho}$: $$q_{0}\in B^{{\frac{N}{p}}}_{p,1},\;u_{0}\in B^{\frac{N}{p_{1}}-1}_{p_{1},1}\;\;\mbox{and}\;\;f\in L^{1}_{loc}({\mathbb{R}}^{+},\in B^{\frac{N}{p_{1}}-1}_{p_{1},1})$$ with $(p,p_{1})\in [1,+\infty[$ good chosen.\
Concerning the global existence of strong 