On the Muskat flow

TL;DR

This paper studies the geometric evolution of two fluids separated by a free interface in a porous medium, analyzing stability, equilibria, and long-term behavior with and without phase transition.

Contribution

It introduces a geometric model for Muskat flow that accounts for non-local curvature effects and proves well-posedness, stability, and global convergence of solutions.

Findings

Models are volume preserving and surface area reducing.

All equilibria are characterized and their stability analyzed.

Solutions exist globally and converge exponentially to equilibrium.

Abstract

Of concern is the motion of two fluids separated by a free interface in a porous medium, where the velocities are given by Darcy's law. We consider the case with and without phase transition. It is shown that the resulting models can be understood as purely geometric evolution laws, where the motion of the separating interface depends in a non-local way on the mean curvature. It turns out that the models are volume preserving and surface area reducing, the latter property giving rise to a Lyapunov function. We show well-posedness of the models, characterize all equilibria, and study the dynamic stability of the equilibria. Lastly, we show that solutions which do not develop singularities exist globally and converge exponentially fast to an equilibrium.