Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain

A priori bounds were derived for the flow in a bounded domain for the viscous-porous interfacing fluids. We assumed that the viscous fluid was slow in Ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega _{1}$\end{document}, which was governed by the Boussinesq equations. For a porous medium in Ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega _{2}$\end{document}, we supposed that the flow satisfied the Darcy equations. With the aid of these a priori bounds we were able to demonstrate the result of the continuous dependence type for the Boussinesq coefficient λ. Following the method of a first-order differential inequality, we can further obtain the result that the solution depends continuously on the interface boundary coefficient α. These results showed that the structural stability is valid for the interfacing problem.