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

## TL;DR

This paper proves the well-posedness of a coupled micro-macro parabolic-elliptic system modeling pressures in a porous medium and establishes a local stability estimate for an inverse Robin problem to identify interfacial transfer coefficients.

## Contribution

It introduces a rigorous mathematical framework for the well-posedness and inverse stability of a multiscale coupled system in porous media, using advanced energy and regularity techniques.

## Key findings

- Well-posedness of the coupled system established.
- Local stability estimate for the inverse Robin problem proved.
- Methodology applicable to micro-macro interfacial problems in porous media.

## 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 Fr\'echet differentiability of the solution and the structure of the inverse stability estimate.