density belongs to larger spaces $B^{{\frac{N}{p}}}_{p,1}$ with $p\in[1,+\infty[$. In the present paper, we address the question of global existence of strong solution in the critical functional framework under the assumption that the initial density belongs to critical Besov space with a index of integrability different of this of the velocity. To do that, as in [@H2] we introduce a new variable in high frequencies than the velocity that we call effective velocity in the goal to *kill* the relation of coupling between the velocity and the pressure. We observe that this new notion of effective velocity allow us easily to get as R. Danchin in [@DG] a $L^{1}$ decay on $q$. However this new variable is interesting only in high frequencies, indeed in low frequencies the term ${\nabla}P(\rho)$ is small in Fourier analysis. Moreover in the low frequency regime, the first order terms predominate and the viscous term ${\Delta}u$ may be neglected in Fourier analysis, so that (\[0.1\]) has to be treat by means of hyperbolic energy methods (more particularly the velocity verifies in some way a wave equation). This implies that we can treat the low regime only in space construct on $L^{2}$, it is classical that the hyperbolic 