of $n_{\epsilon}$, ${n_1}_{\epsilon}$, ${n_2}_{\epsilon}$ and ${p}_{\epsilon}$**]{}.

Integrating on $(-L,L)$ and using the nonnegativity of the densities from Lemma \[lem:nonneg2\] as well as the Neumann boundary conditions, we deduce $$\frac{d}{dt} \| n_{\epsilon} \|_{L^1(-L,L)} \leq G_m \|n_\epsilon\|_{L^1(-L,L)}.$$ Integrating in time, we deduce $$\| n_{\epsilon} \|_{L^1(-L,L)} \leq e^{G_mt} \| n_{\epsilon}^{\mbox{\scriptsize ini}}\|_{L^1(-L,L)}.$$ Since ${n_1}_{\epsilon} \geq 0$ and ${n_2}_{\epsilon} \geq 0$, we deduce the uniform bounds on $\| {n_1}_{\epsilon}\|_{L^1(-L,L)}$ and on $\| {n_2}_{\epsilon}\|_{L^1(-L,L)}$.

From the relation , we deduce $p_{\epsilon}= n_{\epsilon} (\epsilon+p_{\epsilon})$. Moreover, the bound $p_{\epsilon} \leq P_M:=\max (P^1_M,P^2_M) $ implies $$\| p_{\epsilon}\|_{L^1(-L,L)} \leq (\epsilon+P_M)\int_{-L}^L |n_{\epsilon}|\,dx \leq Ce^{G_mt} \| n^{\mbox{\scriptsize ini}}_{\epsilon}\|_{L^1(-L,L)}.$$

[**$L^{1}$ estimates on the $x$ derivatives**]{}.

Recalling , we can refomulate by $$\label{eq:nn2}
\partial_t {n}_{\epsilon} - \partial_{xx} H({n}_{\epsilon}) = {n}_{\epsilon} G(p_{\epsilon},t,x)$$ with $G(p,t,x)= G_1(p) \mathbf{1}_{x \leq  \zeta_{\epsilon}(t) }+G_2(p)\mathbf{1}_{x \geq  \zeta_{\epsilon}(t) }$. The space derivative of this growth function is given by, $$\partial_{x} G(p,t,x) = (G_1(p)-G_2(p)) \delta_{ x= \zeta_{\epsilon}(t)} +G_1'(p) \partial_{x} p \mathbf{1}_{ x \leq  \zeta_{\epsilon}(t) } + G_2'(p) \partial_{x} p \mathbf{1}_{ x \geq  \zeta_{\epsilon}(t) }.$$ We derive with respect to $x$, $$\begin{aligned}
\partial_t \partial_{x} n_{\epsilon} -\partial_{xx} (\partial_{x} H(n_{\epsilon})) =\ & \partial_x n_{\epsilon} G(p_{\epsilon},t,x) + n_{\epsilon} (G_1(p_{\epsilon})-G_2(p_{\epsilon})) \delta_{ x= \zeta_{\epsilon}(t)}  \\
&+ n_{\epsilon}(G_1'(p_{\epsilon})\mathbf{1}_{  x \leq  \zeta_{\epsilon}(t)} +G_2'(p_{\epsilon}) \mathbf{1}_{ x \geq  \zeta_{\epsilon}(t)}) \partial_{x} p_{\epsilon}.\end{aligned}$$ We multiply 