have canceled the coupling between $v_{1}$ and the density, we next verify easily that we have a control Lipschitz of the gradient of $u$ (it is crucial to estimate the density by the mass equation). In the previous theorem \[theo1\], we have as in [@CMZ] very big restrictions on $p$ ($p<\max(4,2N)$) because the behavior in low frequencies. At the difference with the results of strong solutions in finite time (see [@H2]), we can not choose $p$ arbitrarly big. To overcome this difficulty, we need to add some additional conditions on $(q_{0},u_{0})$ in low frequencies as in [@CD] to avoid these restrictions in the use of the paraproduct laws. We obtain then the following corollary:

Let $P$ a suitably smooth function of the density with $P^{'}(1)>0$, $f\in\widetilde{L}^{1}(\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1}\cap B^{0}_{2,r})$ and $1\leq p_{1}\leq p<+\infty$ such that $\frac{1}{p_{1}}\leq\frac{1}{N}+\frac{1}{p}$. Assume that $u_{0}\in \widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1}\cap B^{0}_{2,r}$, $f\in L^{1}_{loc}({\mathbb{R}}^{+},\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1})$ and $q_{0}\in \widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1}\cap B^{0,1}_{2,r}$ with $r=+\infty$ if $N\geq 3$ and $r=1$ if $N=2$.Then there exists a constant ${\varepsilon}_{0}$ such that if: $$\|q_{0}\|_{\widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1}\cap B^{0,1}_{2,r}}+\|u_{0}\|_{\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1}\cap B^{0}_{2,r}}+\|f\|_{\widetilde{L}^{1}(\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1}\cap B^{0}_{2,r})}\leq{\varepsilon}_{0},$$ then if $\frac{1}{p}+\frac{1}{p_{1}}>\frac{1}{N}$, there exists a global solution $(q,u)$ for system (\[0.1\]) with $1+q$ bounded away from zero and, $$\begin{aligned}
&q\in \widetilde{C}({\mathbb{R}},\widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1}\cap B^{0,1}_{2,r})\cap\widetilde{L}^{1}({\mathbb{R}},
\widetilde{B}^{{\frac{N}{2}}+1,\frac{N}{p}}_{2,p,1}\cap B^{2,1}_{2,r})\;\;\;\mbox{and}\\
&\hspace{2cm}\;u\in \widetilde{C}\big({\mathbb{R}};(\widetilde{B}^{{\frac{N}{2}}-1,\frac{N}{p_{1}}-1}_{2,p_{1},1}+\widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1})\cap B^{0}_{2,r}\big)
\cap\widetilde{L}^{1}({\mathbb{R}},\widetilde{B}^{{\frac{N}{2}}+1,{\frac{N}{p}}+1}_{2,p,1}\cap B^{2}_{2,r}).
\end{aligned}$$ Moreover this solution 