limits denoted $n_{10}$, $n_{20}$, and $p_0$, respectively. Moreover, due to the uniform estimate on $(\partial_x p_\epsilon)_\epsilon$ in $L^2(Q_T)$ from Lemma \[lem:L2dp2\], we may extract a subsequence, still denoted $(\partial_x p_\epsilon)_\epsilon$, which converges weakly in $L^2(Q_T)$ towards $\partial_x p_0$. Passing to the limit in the uniform estimates of Lemma \[lem:estim\] gives and ${n_1}_0, {n_2}_0, n_0, p_0$ belongs to $BV(Q_T)$.

Then, we recall that $$\partial_t n_\epsilon - \partial_{xx} (p_\epsilon - \epsilon \ln(p_\epsilon + \epsilon)) = {n_1}_\epsilon G_1(p_\epsilon)+{n_2}_\epsilon G_2(p_\epsilon).$$ From the uniform bounds on $p_\epsilon$, we get, $$\epsilon \ln \epsilon \leq \epsilon \ln(p_\epsilon + \epsilon) \leq \epsilon \ln(P_M + \epsilon).$$ Thus, the term in the Laplacian converges strongly to $p_0$. Then, thanks to the strong convergence of $n_\epsilon$ and $p_\epsilon$, we deduce that in the sense of distributions $$\partial_t n_0 - \partial_{xx} p_0 = {n_1}_0 G_1(p_0)+{n_2}_0 G_2(p_0).$$

Moreover, let $\phi\in W^{1,\alpha}(Q_T)$ with $\phi(T,x)=0$ ($\alpha>2$) be a test function. We multiply equation by $\phi$ and integrate using the Neumann boundary conditions, we get $$\begin{aligned}
-\int_0^T\int_{-L}^L {n_1}_\epsilon \partial_t \phi \,dtdx
  - \int_{-L}^L {n_1}_\epsilon^{ini}(x) \phi(0,x)\,dx + \int_0^T\int_{-L}^L {n_1}_\epsilon \partial_x p_\epsilon \partial_x \phi \,dxdt & \\
= \int_0^T\int_{-L}^L {n_1}_\epsilon G_1(p_\epsilon) \phi \,dxdt.&\end{aligned}$$ Due to the strong convergence of ${n_1}_\epsilon$ and $p_\epsilon$, we can pass easily to the 