Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system

Abstract

We establish the well-posedness of a coupled micro–macro parabolic–elliptic system modeling the interplay between two pressures in a gas–liquid mixture close to equilibrium that is filling a porous media with distributed microstructures. Additionally, we prove a local stability estimate for the inverse micro–macro Robin problem, potentially useful in identifying quantitatively a micro–macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue we use two-scale energy estimates and two-scale regularity/compactness arguments cast in the Schauder’s fixed point theorem. A number of auxiliary problems, regularity, and scaling arguments are used in ensuring the suitable Fréchet differentiability of the solution and the structure of the inverse stability estimate.