Global-in-time estimates for the 2D one-phase Muskat problem with contact points

TL;DR

This paper establishes global-in-time estimates for the 2D Muskat problem with contact points, using Darcy's law and Sobolev space analysis, extending previous work on more regular viscous flows.

Contribution

It provides the first global-in-time a priori estimates for the 2D Muskat problem with contact points using a singular Darcy flow model.

Findings

Proved global-in-time a priori estimates for solutions near equilibrium.

Analyzed the problem in non-weighted Sobolev spaces without contact angle restrictions.

Extended previous regular viscous flow results to Darcy's law setting.

Abstract

In this paper, we study the dynamics of a two-dimensional viscous fluid evolving through a porous medium or a Hele-Shaw cell, driven by gravity and surface tension. A key feature of this study is that the fluid is confined within a vessel with vertical walls and below a dry region. Consequently, the dynamics of the contact points between the vessel, the fluid and the dry region are inherently coupled with the surface evolution. A similar contact scenario was recently analyzed for more regular viscous flows, modeled by the Stokes [GuoTice2018] and Navier-Stokes [GuoTice2024] equations. Here, we adopt the same framework but use the more singular Darcy's law for modeling the flow. We prove global-in-time a priori estimates for solutions initially close to equilibrium. Taking advantage of the Neumann problem solved by the velocity potential, the analysis is carried out in non-weighted $L^2$-based Sobolev spaces and without imposing restrictions on the contact angles.