On an asymptotic model for free boundary Darcy flow in porous media

TL;DR

This paper rigorously analyzes an asymptotic model for free boundary Darcy flow in porous media, establishing well-posedness and decay properties in critical spaces for low amplitude, large wavelength scenarios.

Contribution

It provides the first rigorous mathematical proof of well-posedness and decay behavior for this specific asymptotic Darcy flow model.

Findings

Proved well-posedness in critical spaces.

Established decay of solutions towards equilibrium.

Analyzed low amplitude, large wavelength regime.

Abstract

We provide a rigorous mathematical study of an asymptotic model describing Darcy flow with free boundary in a low amplitude/large wavelength approximation. In particular, we prove several well-posedness results in critical spaces. Furthermore, we also study how the solution decays towards the flat equilibrium.