$$-N\min(\frac{1}{p_{1}},\frac{1}{p^{'}})<s<1+\frac{N}{p_{1}}.$$ There exists a constant $C$ depending only on $N$, $p$, $p_{1}$, $r$ and $s$ such that for all $a\in L^{\infty}([0,T],B^{\sigma}_{p,r})$ of $({\cal H})$ with initial data $a_{0}$ in $B^{s}_{p,r}$ and $g\in L^{1}([0,T], B^{s}_{p,r})$, we have for a.e $t\in[0,T]$: $$\|q\|_{\widetilde{L}^{\infty}_{t}(B^{s}_{p,r})}+\|q\|_{\widetilde{L}^{1}_{t}(B^{s}_{p,r})}\leq e^{CU(t)}\big(\|q_{0}\|_{B^{s}_{p,r}}+\int^{t}_{0}e^{-CU(\tau)}
\|F(\tau)\|_{B^{s}_{p,r}}d\tau\big),
\label{20}$$ with: $U(t)=\int^{t}_{0}\|{\nabla}u(\tau)\|_{B^{\frac{N}{p_{1}}}_{p_{1},\infty}\cap L^{\infty}}d\tau$. \[transport2\]

[**Proof:**]{}Applying ${\Delta}_{l}$ to $({\cal H})$ yields: $${\partial}_{t}{\Delta}_{l}q+u\cdot{\nabla}{\Delta}_{l}q+\alpha{\Delta}_{l}q=R_{l}+{\Delta}_{l}F,$$ with $R_{l}=[u\cdot{\nabla},{\Delta}_{l}]q$. Multiplying by ${\Delta}_{l}a|{\Delta}_{l}a|^{p-2}$ then performing a time integration, we easily get: $$\begin{aligned}
&\|{\Delta}_{l}q(t)\|_{L^{p}}+\alpha\int^{t}_{0}\|{\Delta}_{l}q(s)\|_{L^{p}}ds\leq\|{\Delta}_{l}q_{0}\|_{L^{p}}
+\int^{t}_{0}\big(\|R_{l}\|_{L^{p}}+\frac{1}{p}\|{\rm div}u\|_{L^{\infty}}\|{\Delta}_{l}q\|_{L^{p}}\\
&\hspace{11,5cm}+\|{\Delta}_{l}F\|_{L^{p}}\big)d\tau.
\end{aligned}$$ Next the term $\|R_{l}\|_{L^{p}}$may be bounded according to lemma \[alemme2\] in appendix. We get then: $$\begin{aligned}
&\|q\|_{\widetilde{L}^{\infty}_{t}(B^{s}_{p,r})}+\alpha\|q\|_{\widetilde{L}^{1}_{t}(B^{s}_{p,r})}ds\leq\|{\Delta}_{l}q_{0}\|_{B^{s}_{p,r}}
+\int^{t}_{0}\big(\|F(\tau)\|_{B^{s}_{p,r}}+C U^{'}(\tau)\|q\|_{\widetilde{L}^{\infty}_{t}(B^{s}_{p,r})}\big)d\tau.
\end{aligned}$$ We end up with Gronwall lemma by letting $X(t)=\|q\|_{\widetilde{L}^{\infty}_{t}(B^{s}_{p,r})}+\alpha\|q\|_{\widetilde{L}^{1}_{t}(B^{s}_{p,r})}$.

The proof of theorem \[theo1\] {#section4}
==============================

Strategy of the proof {#section41}
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To improve the results of Danchin in [@DG], Charve and Danchin in [@CD] and Chen et al in [@CMZ], it is crucial to kill the coupling between the velocity and the pressure which exists in these works . In this goal, we need to integrate the pressure term in the study of the linearized equation of the momentum equation as in [@H2]. For making, we will try to express the gradient of the pressure as a Laplacian term, so we set for $\bar{\rho}>0$ a constant 