$(1-n_0) p_0=0$. We consider the domain $\Omega_0(t)=\{x \in (-L,L), p_0(x,t)>0\}$, then, from the latter identity, $n_0 = 1$ on $\Omega_0$. Moreover, from the segregation property, we have ${n_1}_{\epsilon}{n_2}_{\epsilon}=0$ when the two densities are initially segregated. Passing to the limit $\epsilon\to 0$ into this relation implies ${n_1}_{0} {n_2}_{0}=0$. Then we may split $\Omega_0(t)$ into two disjoint sets $\Omega_1(t)=\{x \in (-L,L),{n_1}_{0}(x,t)=1\}$ and $\Omega_2(t)=\{x\in (-L,L),{n_2}_{0}(x,t)=1\}$. Formally, it is not difficult to deduce from that when $\epsilon\to 0$, we expect to have the relation $$-p_0^2 \partial_{xx} p_0 = \left\{
      \begin{aligned}
        p_0^2 G_1(p_0) \text{ on } \Omega_1(t),\\
        p_0^2 G_2(p_0) \text{ on } \Omega_2(t).\\
      \end{aligned}
    \right.$$ Then we obtain a free boundary problem of Hele-Shaw type: On $\Omega_1(t)$, we have ${n_1}_{0}=1$ and $-\partial_{xx} p_0 = G_1(p_0)$, on $\Omega_2(t)$, we have ${n_2}_{0}=1$ and $-\partial_{xx} p_0 = G_2(p_0)$.

The outline of the paper is the following. In Section \[sec:mainresult\] we expose the main results of this paper, which are the convergence of the continuous model - when $\epsilon \rightarrow 0$ to a Hele-Shaw free boundary model, and uniqueness for this limiting model. Section \[sec:proof\] is devoted to the proof of these main results. The proof on the convergence relies on some a priori estimate and compactness techniques. We 