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arXiv · 2502.19286

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

Edoardo Bocchi·Ángel Castro·Francisco Gancedo
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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.

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Edoardo Bocchi, Ángel Castro, Francisco Gancedo. 2025-02-26. Global-in-time estimates for the 2D one-phase Muskat problem with contact points. https://arxiv.org/abs/2502.19286

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