by $ sign(\partial_{x} n_{\epsilon})=sign(\partial_{x} p_{\epsilon})$ and use the Kato inequality, $$\begin{aligned}
\partial_t |\partial_{x} n_{\epsilon}| -\partial_{xx} (|\partial_{x}H(n_{\epsilon})|) \leq \ & |\partial_x n_{\epsilon}| G(p_{\epsilon},t,x) \\
& + n_{\epsilon} (G_1(p_{\epsilon})-G_2(p_{\epsilon})) \delta_{ x= \zeta_{\epsilon}(t)} sign(\partial_{x_i} n_{\epsilon})\\
&+n_{\epsilon}(G_1'(p_{\epsilon})\mathbf{1}_{x \leq  \zeta_{\epsilon}(t)} +G_2'(p_{\epsilon}) \mathbf{1}_{ x \geq  \zeta_{\epsilon}(t)}) |\partial_{x} p_{\epsilon}|.\end{aligned}$$ We integrate in space on $(-L,L)$. Using the fact that $\max_{[0,P^1_M]} G_1'  \leq -\gamma <0$ and $\max_{[0,P^2_M]} G_2'  \leq -\gamma <0$ (see ) and that $\partial_x H(n_{\epsilon}) (\pm L,t)= H'(n_{\epsilon}) \partial_x n_{\epsilon} (\pm L,t) = 0$, $$\begin{aligned}
\partial_t \int_{-L}^L |\partial_{x} n_{\epsilon}|\,dx \leq \ & G_m\int_{-L}^L|\partial_x n_{\epsilon}|\,dx - \gamma \int_{-L}^L n_{\epsilon} |\partial_{x} p_{\epsilon}|\,dx \\
&+ n_{\epsilon}(t, \zeta_{\epsilon}(t)) |G_1(p_{\epsilon}(t, \zeta_{\epsilon}(t))-G_2(p_{\epsilon}(t, \zeta_{\epsilon}(t))| .\end{aligned}$$ Using Gronwall’s lemma and the uniform bound on $n_\epsilon$ and $G_1$ and $G_2$ (see ), we deduce that, for all $t>0$, $$\label{eq:estimdxn2} 
\| \partial_{x} n_{\epsilon}(t) \|_{L^1(-L,L)} + \gamma \int_0^t \int_{-L}^L n_{\epsilon} |\partial_{x} p_{\epsilon}|\,dxds  \leq C e^{G_mt}  \left(\| \partial_{x} n_{\epsilon}^{\mbox{\scriptsize ini}} \|_{L^1(-L,L)} + 1\right) .$$ This conclude the proof for the estimate on $\partial_x n_{\epsilon}$. Then, $$\| \partial_{x} p_{\epsilon}\|_{L^1(-L,L)} = \int_{-L}^L |\partial_{x} p_{\epsilon}|\,dx = \int_{-L}^L\frac{\epsilon}{(1-n_{\epsilon})^2}| \partial_{x} n_{\epsilon}|\,dx.$$ We split the latter integral in two: either $n_{\epsilon}\leq 1/2$ and then $\frac{\epsilon}{(1-n_{\epsilon})^2} \leq C$; either $n_{\epsilon}\geq 1/2$, $$\begin{aligned}
  \| \partial_{x} p_{\epsilon}\|_{L^1(-L,L)}  &\leq  C \int_{n_{\epsilon}\leq 1/2} |\partial_{x} n_{\epsilon}|\,dx + \int_{n_{\epsilon}\geq 1/2} |\partial_{x} 