state: $${\rm div}v=P(\rho)-P(\bar{\rho}).$$ Let ${\cal E}$ the fundamental solution of the Laplace operator. $$$$ We will set in the sequel: $v={\nabla}{\cal E}*\big(P(\rho)-P(\bar{\rho})\big)={\nabla}\big({\cal E}*[P(\rho)-P(\bar{\rho})]\big)$ ( $*$ here means the operator of convolution). We verify next that: $$\begin{aligned}
{\nabla}{\rm div}v={\nabla}{\Delta}\big({\cal E}*[P(\rho)-P(\bar{\rho})]\big)={\Delta}{\nabla}\big({\cal E}*[P(\rho)-P(\bar{\rho})]\big)={\Delta}v={\nabla}P(\rho).
\end{aligned}$$ By this way we can now rewrite the momentum equation of (\[0.6\]). We obtain the following equation where we have set $\nu=2\mu+\lambda$: $${\partial}_{t}u+u\cdot {\nabla}u-\frac{\mu}{\rho}{\Delta}\big(u-\frac{1}{\nu}v\big)-\frac{\lambda+\mu}{\rho}{\nabla}{\rm div}\big(u-\frac{1}{\nu}v\big)=f.$$ We want now calculate ${\partial}_{t}v$, by the transport equation we get: $${\partial}_{t}v={\nabla}{\cal E}*{\partial}_{t}P(\rho)=-{\nabla}{\cal E}*\big(P^{'}(\rho){\rm div}(\rho u)\big).$$ We have finally: $${\Delta}({\partial}_{t}F)=-P^{'}(\rho){\rm div}(\rho u).$$

To simplify the notation, we will note in the sequel $${\nabla}{\cal E}*\big(P^{'}(\rho){\rm div}(\rho u)\big)={\nabla}({\Delta})^{-1}\big(P^{'}(\rho){\rm div}(\rho u)\big).$$

Finally we can now rewrite the system (\[0.6\]) as follows: $$\begin{cases}
\begin{aligned}
&{\partial}_{t}q+(v_{1}+\frac{1}{\nu}v)\cdot{\nabla}q+\frac{1}{\nu}(1+q)(P(\rho)-P(1))=-(1+q){\rm div}v_{1},\\
&{\partial}_{t}v_{1}-\frac{1}{1+q}{\cal A}v_{1}=f-u\cdot{\nabla}u+\frac{1}{\nu}{\nabla}({\Delta})^{-1}\big(P^{'}(\rho){\rm div}(\rho u)\big),\\
&q_{/ t=0}=a_{0},\;(v_{1})_{/ t=0}=(v_{1})_{0}.
\end{aligned}
\end{cases}
\label{0.7}$$ where $v_{1}=u-\frac{1}{\nu}v$ is called the effective velocity. In the sequel we will study this system by exctracting some uniform bounds in Besov spaces on $(q,v_{1})$. The advantage of the system (\[0.7\]) is that we have *kill* the coupling between $v_{1}$ and a term of pressure. Indeed in the works [@CD] and [@CMZ], the pressure was included in the study of the linear system, it means mandatory a coupling between the density and 