Sharp pressure estimates for the Navier-Stokes system in thin porous media

A relevant problem for applications is to model the behavior of Newtonian fluids through thin porous media, which is a domain with small thickness $\epsilon$ and perforated by periodically distributed cylinders of size and period $\epsilon^\delta$, with $\delta>0$. Depending on the relation between thickness and the size of the cylinders, it was introduced in (Fabricius et al., Transp. Porous Media, 115, 473-493, 2016), (Anguiano and Su\'arez-Grau, Z. Angew. Math. Phys., 68:45, 2017) and (Anguiano and Su\'arez-Grau, Mediterr. J. Math., 15:45, 2018) that there exist three regimes depending on the value of $\delta$: $\delta\in (0,1)$, $\delta=1$ and $\delta>1$. In each regime, the asymptotic behavior of the fluid is governed by a lower-dimensional Darcy's law. In previous studies, the Reynolds number is considered to be of order one and so, the question that arises is for what range of values of the Reynolds number the lower-dimensional Darcy laws are still valid in each regime, which represents the main the goal of this paper. In this sense, considering a fluid governed by the Navier-Stokes system and assuming the Reynolds number written in terms of the thickness $\epsilon$, we prove that, for each regime, there exists a critical Reynolds number $Re_c$ such that for every Reynolds number $Re$ with order smaller or equal than $Re_c$, the lower-dimensional Darcy law is still valid. On the contrary, for Reynolds numbers $Re$ greater than $Re_c$, the inertial term of the Navier-Stokes system has to be taken into account in the asymptotic behavior and so, the Darcy law is not valid.