A fluid-solid interaction problem in porous media

TL;DR

This paper derives and analyzes asymptotic models for fluid-solid interactions in porous media, focusing on free boundary dynamics and lubrication regimes, with proven well-posedness in functional spaces.

Contribution

It introduces new nonlocal and lubrication-type equations for elastic Muskat problems, extending understanding of free boundary fluid-structure interactions.

Findings

Derived nonlocal evolution equations for free boundary dynamics.

Reformulated kinematic condition into flux form in thin-film regime.

Proved well-posedness of models in Wiener spaces.

Abstract

In this work, we derive asymptotic interface models for an elastic Muskat free boundary problem describing Darcy flow beneath an elastic membrane. In a weakly nonlinear regime of small interface steepness, we obtain nonlocal evolution equations that capture the free-boundary dynamics up to quadratic order. In the long-wave thin-film regime, we rewrite the kinematic condition in flux form, flatten the moving domain, and derive a lubrication-type equation. Moreover, we establish well-posedness for these models in suitable Wiener spaces.