are able to obtain a priori estimates on the densities, the pressure and their spatial derivatives. Compactness in time is deduced thanks to the Aubin-Lions theorem. The proof of convergence follows from these new estimates. However, the lack of estimates on the time derivative makes obtaining the complementary relation difficult, then we are not able to recover the usual relation but the one stated in which may be seen as an integral in time of the usual one as explained in the above remark.

To complete the result on the asymptotic limit of the model, an uniqueness result for the Hele-Shaw free boundary model for two populations is provided in Proposition \[TH2\] in §\[sec:uniq\]. The proof of this uniqueness result for the limiting problem is based on Hilbert’s duality method.

Proof of the main results {#sec:proof}
=========================

This section is devoted to the proof of Theorem \[TH1\], whereas in Section \[sec:uniq\] the uniqueness of the solution to the Hele Shaw system is established. We first establish some a priori estimates.

A priori estimates
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### Nonnegativity principle

The following Lemma establishes the nonnegativity of the densities.

\[lem:nonneg2\] Let $({n_1}_\epsilon,{n_2}_\epsilon,p_\epsilon)$ be a solution to and such that ${n_1}_\epsilon^{\mbox{\scriptsize ini}}\geq 0$, ${n_2}_\epsilon^{\mbox{\scriptsize 