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

Edoardo Bocchi, Ángel Castro, Francisco Gancedo

TL;DR

This work analyzes the global-in-time behavior of the 2D Muskat problem with contact points, modeled in a vessel with vertical walls and a dry region, using Darcy's law. The authors develop a fixed-domain framework via a velocity potential Neumann problem and a carefully constructed diffeomorphism about a stationary state, enabling a bootstrap to higher regularity through elliptic estimates in domains with corners. They introduce an improved energy-dissipation structure that closes the nonlinear estimates and proves exponential decay of the parallel energy for small perturbations around the stationary profile. The analysis extends techniques from Stokes/Navier–Stokes contact-point problems to the more singular Darcy flow, handling arbitrary contact angles in (0,π) without weighting and leveraging corner spectral analysis to control the elliptic problems. The results pave the way for global well-posedness and stability results for Muskat-type interfaces with capillarity and moving contact lines, with a companion paper addressing local well-posedness.

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.

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

TL;DR

This work analyzes the global-in-time behavior of the 2D Muskat problem with contact points, modeled in a vessel with vertical walls and a dry region, using Darcy's law. The authors develop a fixed-domain framework via a velocity potential Neumann problem and a carefully constructed diffeomorphism about a stationary state, enabling a bootstrap to higher regularity through elliptic estimates in domains with corners. They introduce an improved energy-dissipation structure that closes the nonlinear estimates and proves exponential decay of the parallel energy for small perturbations around the stationary profile. The analysis extends techniques from Stokes/Navier–Stokes contact-point problems to the more singular Darcy flow, handling arbitrary contact angles in (0,π) without weighting and leveraging corner spectral analysis to control the elliptic problems. The results pave the way for global well-posedness and stability results for Muskat-type interfaces with capillarity and moving contact lines, with a companion paper addressing local well-posedness.

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 -based Sobolev spaces and without imposing restrictions on the contact angles.

Paper Structure

This paper contains 22 sections, 22 theorems, 258 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contact points
  3. Main results
  4. Previous results and main novelties
  5. General notation
  6. Outline of the paper
  7. Potential and fixed-boundary formulations
  8. Potential formulation
  9. Fixed-boundary formulation
  10. Basic energy-dissipation equalities
  11. Energies and dissipations
  12. Additional dissipation
  13. Control of the $L^2$-norm of $\partial_t^j \Phi$
  14. Higher regularity for $\partial_t^j \eta$
  15. Elliptic estimates in $\Omega_s$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $h(t,x)=h_s(x)+\eta(t,x)$ and $h_s(x)$ be, respectively, the surface profiles of the dynamical and stationary solution to incompre-dyn-law. Assume that $h$ and $h_s$ have equal mean in $\mathcal{I}$. Then, for any $t\in [0,T]$ with $T>0$, the following bound holds: provided that $\mathcal{E}(0)$ is small enough. Above, the energy term $\mathcal{E}(t)$ is given by for any $0<\delta\leq 1$, an

Figures (3)

  • Figure 1: Configuration of the vessel.
  • Figure 2: A possible equilibrium configuration.
  • Figure 3: Plots of the different shapes of $h_s$ with zero mean in $\mathcal{I}$: concave when $\llbracket \gamma \rrbracket<0$, convex when $\llbracket \gamma \rrbracket>0$ and flat when $\llbracket \gamma \rrbracket=0$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • ...and 36 more