system are ill-posed in general $L^{p}$ spaces. So as in [@CMZ] and [@CD], the system has to be handled differently in low and high frequencies. In short, we will use the analysis of R. Danchin in [@DG] in low frequencies and the introduction of this new variable the effctive velocity introduced in [@H2] in high frequencies. To simplify the notation, we assume from now on that $\bar{\rho}=1$. Hence as long as $\rho$ does not vanish, the equations for ($q=\rho-1$,$u$) read: $$\begin{cases}
\begin{aligned}
&{\partial}_{t}q+u\cdot{\nabla}q=-(1+q){\rm div}u,\\
&{\partial}_{t}u+u\cdot{\nabla}u-\frac{1}{1+q}{\cal A}u+{\nabla}P(1+q)=f,
\end{aligned}
\end{cases}
\label{0.6}$$ In the sequel we will note ${\cal A}=\mu{\Delta}+(\lambda+\mu){\nabla}{\rm div}$ and where $g$ is a smooth function which may be computed from the pressure function $P$. One can now state our main result.

Let $P$ a suitably smooth function of the density such that $P^{'}(1)>0$, $f\in\widetilde{L}^{1}(\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1})$ and $1\leq p_{1}\leq p<+\infty$ such that $\frac{1}{p_{1}}\leq\frac{1}{N}+\frac{1}{p}$. Assume that $u_{0}\in \widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1}$, $f\in L^{1}_{loc}({\mathbb{R}}^{+},\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1})$ and $q_{0}\in \widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1}$.Then there exists a constant ${\varepsilon}_{0}$ such that if: $$\|q_{0}\|_{\widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1}}+\|u_{0}\|_{\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1}}+\|f\|_{\widetilde{L}^{1}(\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1})}\leq{\varepsilon}_{0},$$ then if $\frac{1}{p}+\frac{1}{p_{1}}>\frac{1}{N}$, $p<\max(4,N)$ and $\frac{1}{2}\leq\frac{1}{p}+\frac{1}{p_{1}}$ there exists a global solution $(q,u)$ for system (\[0.1\]) with $1+q$ bounded away from zero and, $$\begin{aligned}
&q\in \widetilde{C}({\mathbb{R}},\widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1})\cap\widetilde{L}^{1}({\mathbb{R}},
\widetilde{B}^{{\frac{N}{2}}+1,\frac{N}{p}}_{2,p,1})\;\;\;\mbox{and}\\
&\hspace{2cm}\;u\in \widetilde{C}({\mathbb{R}};\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1}+\widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1})
\cap\widetilde{L}^{1}({\mathbb{R}},\widetilde{B}^{{\frac{N}{2}}+1,{\frac{N}{p}}+1}_{2,p,1}).
\end{aligned}$$ Moreover this solution is unique if $\frac{2}{N}\leq\frac{1}{p}+\frac{1}{p_{1}}$. \[theo1\]

This theorem is 