$$\int_{-L}^L (1-n_{\epsilon})p_{\epsilon} |\ln(p_{\epsilon})|\,dx \leq C \int_{-L}^L \mathbf{1}_{p_{\epsilon}>0} \,dx \leq 2LC.$$ This concludes the proof.

Proof of theorem 1
------------------

### Convergence

In the last paragraph we have found a priori estimates for the densities and their space derivatives. To use a compactness argument, we need to obtain estimates on the time derivative. To do so, we are going to use the Aubin Lions theorem [@simon].

According to Lemma \[lem:L2dp2\], ${n_1}_{\epsilon}\partial_x p_\epsilon$ and ${n_2}_{\epsilon}\partial_x p_\epsilon$ are in $L^2(Q_T)$. Moreover thanks to Lemma \[lem:estim\], we have that ${n_1}_{\epsilon} G_1(p_{\epsilon})$ and ${n_2}_{\epsilon} G_2(p_{\epsilon})$ are uniformly bounded in $L^\infty([0,T];L^1\cap L^\infty(-L,L))$, so $\partial_t {n_1}_{\epsilon}$ and $\partial_t {n_2}_{\epsilon}$ are uniformly bounded in $ L^2([0,T],W^{-1,2}(-L,L))$. We also have ${n_1}_{\epsilon}$ and ${n_2}_{\epsilon}$ bounded in $L^1([0,T],W^{1,1}(-L,L))$. Since we are working in one dimension, we have the following embeddings $$W^{1,1}(-L,L) \subset L^1(-L,L) \subset W^{-1,2}(-L,L).$$ The Aubin Lions theorem implies that $\{u \in L^1([0,T],W_{loc}^{1,1}(-L,L)); \dot{u} \in L^2([0,T],W^{-1,2}(-L,L))  \}$ is compactly embedded in $L^1([0,T],L^1(-L,L))$. So we can extract strongly converging subsequences ${n_1}_{\epsilon}$ and ${n_2}_{\epsilon}$ in $L^1(Q_T)$. The convergence of the pressure follows from the same kind of computation.

### Limit model

From the above results, up to extraction of subsequences, $(n_{1\epsilon})_\epsilon$, $(n_{2\epsilon})_\epsilon$, and $(p_\epsilon)_\epsilon$ converge strongly in $L^1(Q_T)$ and a.e. towards some 