with $K>0$. To speak roughly under the conditions that two solutions $(\rho,u)$, $(\rho_{1},u_{1})$ check a control $L^{\infty}$ on the density and a control Lipschitz on the velocity, with additional property of regularity on the strong solution $(\rho_{1},u_{1})$ then we obtain $(\rho,u)=(\rho_{1},u_{1})$. D. Hoff use this result to show that the solutions of [@5H2] are unique. We will use this theorem in the sequel, by showing that our solutions verify the hypothesis of D. Hoff in [@5H2]. Finally R. Danchin in [@DG] show for the first time a result of existence of global strong solution close from a stable equilibrium in critical space for the scaling of the system. More precisely the initial data are choose as follows $(q_{0},u_{0})\in (B^{{\frac{N}{2}}}_{2,1}\cap B^{{\frac{N}{2}}-1}_{2,1})\times B^{{\frac{N}{2}}-1}_{2,1}$. The main difficulty is to get estimates on the linearized system where the velocity and the density are coupled via the pressure, and what is crucial in this work is the smoothing effect on the velocity and a $L^{1}$ decay on $\rho-\bar{\rho}$ (this play a necessary role to control the pressure term). In this work, R. Danchin use some astucious inequality of energy on the system in variable Fourier where he has decomposed the space in dyadic shell. This 