Asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones

TL;DR

This paper rigorously analyzes the long-time behavior of Darcy--Boussinesq convection in layered porous media, proving convergence to a sharp-interface model and establishing properties of the global attractor.

Contribution

It provides the first long-time convergence proof of the global attractors and invariant measures for the model as transition zones vanish.

Findings

Global attractors converge semi-continuously to the sharp-interface model.

Global attractors have finite fractal dimension with explicit bounds.

The analysis confirms the model's long-time validity and extends previous finite-time results.

Abstract

We study the asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones in the material properties. As the transition-layer width tends to zero, we prove the upper semi-continuous convergence of the global attractor, invariant measure, and Nusselt number to their counterparts in the limiting sharp-interface model. We also show that the global attractors have finite fractal dimensions, with an explicit upper bound uniform in the transition-layer width. The analysis combines a carefully designed background temperature/contaminant profile together with a novel choice of phase space that ensures global well-posedness of the model and asymptotic compactness of the solution semigroup, and a new interpolation inequality. The phase space is associated with fractional powers of the principal elliptic operator with discontinuous coefficients. These results provide a rigorous long-time validation of the sharp-interface Darcy--Boussinesq model and extend our earlier finite-time convergence theory (H. Dong and X. Wang, SIAM J. Appl. Math. 85 (2025), 1621--1642) to the long-time regime.