is unique if $\frac{2}{N}\leq\frac{1}{p}+\frac{1}{p_{1}}$. \[corollaire1\]

We can observe that when $p$ tends to infinity, we are close to get weak solution with the following initial data $(q_{0},u_{0})$ in $B^{{\frac{N}{2}}-1,0}_{2,\infty,1}\times B^{{\frac{N}{2}}-1,1}_{2,N,1}$. It means that this theorem rely the result of D. Hoff where the initial density is assumed $L^{\infty}$ but where the initial velocity is more regular (it means not critical) and the results of R. Danchin in [@DW].Moreover it is right for general pressure when in the works of D. Hoff the pressure verify $P(\rho)=K\rho$ with $K>0$.

If $r=+\infty$, then we replace above the strong continuity in $B^{s}_{2,r}$ by the weak continuity.

In some some sense, we could consider that the case $p>N$ is not so important because we obtain only the existence of global weak solution as in the works of D. Hoff in [@5H4], [@5H2]. However it stays very interesting, indeed as in the work of F. Charve and R. Danchin in [@CD], we could add some additional condition on the data such $u_{0}\in B^{0}_{N,1}$. In this fact, it would be easy to show some results of persistency as for Navier-Stokes without condition of smallness on $\|u_{0}\|_{B^{0}_{N,1}}$, in particular the fact that $u\in\widetilde{L}^{\infty}(\widetilde{B}^{{\frac{N}{2}}-1,0}_{2,N,1}+\widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{2,p,1})\cap \widetilde{L}^{1}(\widetilde{B}^{{\frac{N}{2}}+1,{\frac{N}{p}}+1}_{2,p,1})$. And in this case, 