the following system: $$\begin{cases}
\begin{aligned}
&{\partial}_{t}\rho+{\rm div}(\rho u)=0,\\
&{\partial}_{t}(\rho u)+{\rm div}(\rho u\otimes u)-{\rm div}(\mu(\rho)D(u))-{\nabla}(\lambda(\rho){\rm div} u)+{\nabla}P(\rho)=\rho f,\\
&(\rho,u)_{/t=0}=(\rho_{0},u_{0}).
\end{aligned}
\end{cases}
\label{0.1}$$ Here $u=u(t,x)\in{\mathbb{R}}^{N}$ stands for the velocity field and $\rho=\rho(t,x)\in{\mathbb{R}}^{+}$ is the density. The pressure $P$ is a suitable smooth function of $\rho$. We denote by $\lambda$ and $\mu$ the two viscosity coefficients of the fluid, which are assumed to satisfy $\mu>0$ and $\lambda+2\mu>0$ (in the sequel to simplify the calculus we will assume the viscosity coefficients as constants). Such a conditions ensures ellipticity for the momentum equation and is satisfied in the physical cases where $\lambda+\frac{2\mu}{N}>0$. We supplement the problem with initial condition $(\rho_{0},u_{0})$ and an outer force $f$. Throughout the paper, we assume that the space variable $x\in{\mathbb{R}}^{N}$ or to the periodic box ${\cal T}^{N}_{a}$ with period $a_{i}$, in the i-th direction. We restrict ourselves the case $N\geq2$.\
The problem of existence of global solution in time for Navier-Stokes equations was addressed in one dimension for smooth enough data by Kazhikov and Shelukin in [@5K], and for discontinuous ones, but still with densities away from zero, by Serre in [@5S] and Hoff in [@5H1]. Those results have been generalized to higher dimension by Matsumura and Nishida in [@MN] 