Global-in-Time Estimates for the 2D One-Phase Muskat Problem with Contact Points

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 (Guo and Tice, Arch Ration Mech Anal 227(2):767–854, 2018) and Navier–Stokes (Guo and Tice, J Eur Math Soc 26(4):1445–1557, 2024) 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.