priori estimates

To show the compactness result we establish a priori estimate on the densities, pressure and their derivatives. We first compute the equation on the total density. As shown earlier ${n_1}_\epsilon$ and ${n_2}_\epsilon$ are respectively weak solutions of and . By summing the two equations we deduce that ${n}_\epsilon$ is a weak solution of . Notice that this equation can be rewritten as, $$\label{eq:nH}
\partial_t n_{\epsilon} -\partial_{xx} H(n_{\epsilon})= {n_1}_{\epsilon} G_1(p_{\epsilon}) + {n_2}_{\epsilon} G_2(p_{\epsilon}),$$ with $H(n)=\int_0^{n} u P'(u)du = P(n)-\epsilon \ln (P(n)+\epsilon) + \epsilon \ln \epsilon$.

We establish the following a priori estimates

\[lem:estim\] Let us assume that and hold. Let $({n_1}_\epsilon,{n_2}_\epsilon,p_\epsilon)$ be a solution to –. Then, for all $T>0$, and $t\in (0,T)$, we have the uniform bounds in $\epsilon\in(0,\epsilon_0)$, $$\begin{aligned}
& {n_1}_{\epsilon}, {n_2}_{\epsilon} \mbox{ in } L^\infty([0,T];L^1\cap L^\infty(-L,L)); \\
& 0\leq p_\epsilon \leq P_M, \qquad 0  < A_0 e^{-g_m t}\leq n_\epsilon(t) \leq \frac{P_M}{P_M+\epsilon} \leq 1.\end{aligned}$$ Moreover, we have that $({n_1}_\epsilon)_\epsilon$ and $({n_2}_\epsilon)_\epsilon$ are uniformly bounded in $L^\infty([0,T],W^{1,1}(-L,L))$ and $(p_\epsilon)_\epsilon$ is uniformly bounded in $L^1([0,T],W^{1,1}(-L,L))$.

[**Comparison principle**]{}.

The usual comparison principle is not true for this system of equations. However we are able to show some comparison between the total density and $n_M$ defined by $n_M= \frac{P_M}{\epsilon+P_M}$ where $P_M$ 