will be used in the following.

Considering this particular solution, we are going to show the incompressible limit $\epsilon \rightarrow 0$ for system -. The main result is the following

\[TH1\] Let $T>0$, $Q_T=(0,T)\times(-L,L)$. Let $G_1$, $G_2$ and $({n_1}^{\mbox{\scriptsize ini}}_{\epsilon})$, $({n_2}^{\mbox{\scriptsize ini}}_{\epsilon})$ satisfy assumptions –. After extraction of subsequences, the densities ${n_1}_{\epsilon}$, ${n_2}_{\epsilon}$ and the pressure $p_{\epsilon}$, solutions defined in -, converge strongly in $L^1(Q_T)$ as $\epsilon \rightarrow 0$ towards the respective limit ${n_1}_0, {n_2}_0 \in L^\infty([0,T];L^1(-L,L))\cap BV(Q_T)$, and $p_0 \in BV(Q_T)\cap L^2([0,T];H^1(-L,L))$. Moreover, these functions satisfy, for all $(t,x)\in Q_T$, $$\begin{aligned}
& 0 \leq {n_1}_0(t,x) \leq 1, \quad 0 \leq {n_2}_0(t,x) \leq 1,  \label{unif0}  \\
&  0 < A_0 e^{- g_m t} \leq n_0(t,x) \leq 1, \quad 0 \leq p_0 \leq P_M, \label{unif0np} \\
& \partial_t n_0 - \partial_{xx} p_0 = {n_1}_0 G_1(p_0)+{n_2}_0 G_1(p_0), \mbox{ in } \mathcal{D}'(Q_T), \label{limitn0}\end{aligned}$$ where $n_0={n_1}_0 +{n_2}_0$, and $$\begin{aligned}
\partial_t {n_1}_0 - \partial_x ( {n_1}_0 \partial_x p_0) = {n_1}_0 G_1(p_0), & \quad \text{ in } \mathcal{D}'(Q_T), \label{limitn1} \\
\partial_t {n_2}_0 - \partial_x ( {n_2}_0 \partial_x p_0) = {n_2}_0 G_2(p_0), & \quad \text{ in } \mathcal{D}'(Q_T), \label{limitn2}\end{aligned}$$ complemented with Neumann boundary conditions $\partial_x p_0(\pm L)  = 0$. Moreover, we have the relations $$\begin{aligned}
\label{n0p02}
(1-n_0)p_0=0,\end{aligned}$$ 