Effective interface laws for Darcy--Boussinesq convection across a vanishing low-permeability layer

Abstract

We study Darcy--Boussinesq convection in a layered porous medium containing a thin layer whose thickness $h_0$ and permeability $K_0$ vanish simultaneously. The competition between these two small parameters is measured by the effective hydraulic conductance $α= \lim \frac{K_0}{h_0} \in [0,\infty]$. We show that the collapsing layer is replaced by an effective interface whose transmission law depends qualitatively on this ratio. When $α= 0$, the limiting interface is impermeable; when $ 0 <α<\infty$, the normal Darcy flux is proportional to the pressure jump, yielding a finite-resistance transmission law; and when $α=\infty$, the pressure is continuous and the interface becomes hydraulically transparent. We establish estimates uniform in the layer thickness and permeability, quantitative convergence of solutions on every finite time interval, and upper semicontinuity of the associated global attractors. The analysis has to accommodate both a degenerating geometry and a loss of coercivity in the pressure equation. In particular, we use a generalized Poincaré inequality for piecewise $H^1$ pressures, a pressure extension across the collapsing layer, and uniform thin-strip estimates. The results provide a rigorous finite- and long-time validation of effective interface models for convection through thin low-permeability interphases.