case of lows frequencies, according the point of view of the Fourier frequencies, the term of pressure is very regular so it does not make problem to consider in the rest. The study of the linear part in this case is crucial to get a gain $L^{1}$ of integrability on the density, and for low frequencies we follow the method of R. Danchin in [@DG]. In [@5H4; @5H5], D. Hoff get global weak solution with a critical regularity on the density in the sense that $\rho_{0}-\bar{\rho}$ is small in $L^{\infty}$. It means that he does not ask any regularity on the initial density which is of this point of view besser than [@DG], [@CD] and [@CMZ]. However the velocity in his case in only $L^{1}$ $log$ Lipschitz, that is why he can not obtain the uniqueness and have only weak solution. In this paper we improve the results of [@5H4; @5H5] by the fact that we get strong solutions, and we will show that it is just enough to ask a arbitrary small ${\varepsilon}$ regularity on the initial density to get uniqueness.\
In [@H2], we improve the results of R. Danchin in [@DL; @DW], in the sense that the initial 