shall also make use of hybrid Besov-spaces. For them, one can prove results analoguous to proposition \[produit1\], we refer to proposition \[hybrid\] in the appendix.

The mass conservation equation {#section3}
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We begin this section by recalling some estimates in Besov spaces for transport and heat equations. For more details, the reader is referred to [@BCD].

Let $1\leq p_{1}\leq p\leq+\infty$, $r\in[1,+\infty]$ and $s\in{\mathbb{R}}$ be such that: $$-N\min(\frac{1}{p_{1}},\frac{1}{p^{'}})<s<1+\frac{N}{p_{1}}.$$ Suppose that $q_{0}\in B^{s}_{p,r}$, $F\in L^{1}(0,T, B^{s}_{p,r})$ and that $q\in
L^{\infty}_{T}(B^{s}_{p,r})\cap
C([0,T];{\cal S}^{'})$ solves the following transport equation: $$\begin{cases}
\begin{aligned}
&{\partial}_{t}q+u\cdot{\nabla}q=F,\\
&q_{\ t=0}=q_{0}.
\end{aligned}
\end{cases}$$ There exists a constant $C$ depending only on $N$, $p$, $p_{1}$, $r$ and $s$ such that , we have for a.e $t\in[0,T]$: $$\|q\|_{\widetilde{L}^{\infty}_{t}(B^{s}_{p,r})}\leq e^{CU(t)}\big(\|q_{0}\|_{B^{s}_{p,r}}+\int^{t}_{0}e^{-CU(\tau)}
\|F(\tau)\|_{B^{s}_{p,r}}d\tau\big),
\label{20}$$ with: $U(t)=\int^{t}_{0}\|{\nabla}u(\tau)\|_{B^{\frac{N}{p_{1}}}_{p_{1},\infty}\cap L^{\infty}}d\tau$. \[transport1\]

We want study now the following problem: $$\begin{cases}
\begin{aligned}
&{\partial}_{t}q+u\cdot{\nabla}q+\alpha q=F,\\
&q_{\ t=0}=q_{0}.
\end{aligned}
\end{cases}
\leqno{({\cal H})}$$ Above $a$ is the unknown function. We assume that $F\in L^{r}(0,T;B^{s}_{p,r})$, that $v$ is time dependent vector-fields with coefficients in $L^{1}(0,T;B^{\frac{N}{p_{1}}+1}_{p_{1},1})$ and $\alpha>0$.\
Indeed we recall that we can rewrite the transport equation on the following form: $${\partial}_{t}q+u\cdot{\nabla}q+(q+1)(P(1+q)-P(1))=-(1+q){\rm div}v_{1},$$ where we refer to the section \[section41\] for the definition of $v_{1}$.

Let $1\leq p_{1}\leq p\leq+\infty$, $r\in[1,+\infty]$ and $s\in{\mathbb{R}}$ be such that: 