the same than Chen et al obtained in [@CMZ]. In particular we have strong restrictions on $p$, it means $p<\max(4,2N)$. This fact is due to the interactions between low and high frequencies in the paraproduct laws. However we obtain this result with a new method which seems more flexible and we will explain why in thecorrolary \[corollaire1\].

It seems possible to improve the theorem \[theo1\] by choosing initial data $q_{0}$ in $B^{{\frac{N}{2}}-1,{\frac{N}{p}}}_{(2,1),(p,\infty)}\cap B^{{\frac{N}{2}}-1,0}_{(2,1),(\infty,1)}$, however some supplementary conditions appear on $p_{1}$ in this case. Here $B^{s_{1},s_{2}}_{(p_{1},r_{1}),(p_{2},r_{2})}$ is a Besov space where the behavior is $B^{s_{1}}_{p_{1},r_{1}}$ and $B^{s_{2}}_{p_{2},r_{2}}$ in high frequencies.

The key to theorem \[theo1\] is to introduce a new variable $v_{1}$ to control the velocity where to avoid the coupling between the density and the velocity, we analyze by a new way the pressure term. More precisely we write the gradient of the pressure as a Laplacian of the variable $v_{1}$, and we introduce this term in the linear part of the momentum equation. We have then a control on $v_{1}$ which can write roughly as $u-{\cal G}P(\rho)$ where ${\cal G}$ is a pseudodifferential operator of order $-1$. We will call $u-{\cal G}P(\rho)$ the effective velocity. By this way, we 