explain in particular why the result is obtained in Besov space with a Lebesgue index $p=2$. In the same time of the redaction of this paper, Q. Chen et al in [@CMZ] and F. Charve and R. Danchin in [@CD] improve the previous result by working in more general Besov space by studying the linear part of the system.\
The goal of this article is to make a connection between the article of D. Hoff [@5H4; @5H5] and those of Q. Chen et al and F. Charve and R. Danchin in [@CD] and [@CMZ]. In fact we extend the results [@CD] and [@CMZ] to the case where the Lebesgue index of Besov space are more general, it means $q_{0}\in B^{{\frac{N}{p}}}_{p,1}$ and $\;u_{0}\in B^{\frac{N}{p_{1}}-1}_{p_{1},1}$ with $p$ and $p_{1}$ good choosen. In [@CD] and [@CMZ], the authors obtain global weak solutions when $p=p_{1}<2N$ and strong solution when $p=p_{1}\leq N$, the restriction on the choice of $p$ come from a very strong coupling between the pressure and the velocity. Indeed in this case the coupling is very strong in high frequencies because of the term of pressure, that’s why we need to integrate completly the pressure term in the linear part. In the 