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arxiv: 2502.19286 · v2 · submitted 2025-02-26 · 🧮 math.AP

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

Edoardo Bocchi , \'Angel Castro , Francisco Gancedo This is my paper

Pith reviewed 2026-05-23 02:02 UTC · model grok-4.3

classification 🧮 math.AP
keywords Muskat problemcontact pointsglobal a priori estimatesDarcy's lawHele-Shaw cellSobolev spacesfree boundary problem
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The pith

Global-in-time a priori estimates hold for the 2D one-phase Muskat problem with contact points when initial data is close to equilibrium.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that solutions to the two-dimensional Muskat problem modeling viscous flow through a porous medium or Hele-Shaw cell remain controlled for all time if they begin sufficiently near the flat equilibrium state. The fluid sits in a vessel with vertical walls, so the contact points where the free surface meets the walls evolve in lockstep with the interface. The authors work directly with the velocity potential satisfying a Neumann boundary-value problem, which lets them close the estimates in ordinary L2-based Sobolev spaces and without any restriction on the contact angles. This mirrors earlier global bounds obtained for the Stokes and Navier-Stokes versions of the same geometry but now applies to the more singular Darcy-law case. A reader cares because such uniform bounds are the necessary first step toward proving long-time existence and asymptotic stability for contact-point free-boundary problems.

Core claim

For initial data close enough to equilibrium, the one-phase Muskat problem in two dimensions with contact points admits global-in-time a priori estimates in non-weighted L2-based Sobolev spaces. The analysis exploits the Neumann problem satisfied by the velocity potential to control the evolution of both the free surface and the contact points without imposing angle restrictions.

What carries the argument

The Neumann problem solved by the velocity potential, used to obtain Sobolev estimates on the interface and contact-point motion.

If this is right

  • Solutions that begin close to equilibrium remain close for all positive times.
  • The contact points cannot detach or cause finite-time singularities under the smallness assumption.
  • The estimates close without any restriction on the size of the contact angles.
  • The same framework that worked for Stokes and Navier-Stokes now extends to the Darcy-law Muskat setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method might adapt to three-dimensional versions of the same contact geometry.
  • Numerical simulations of near-equilibrium data could check whether the derived bounds are close to sharp.
  • The absence of angle restrictions opens the possibility of applying the estimates to physical regimes where contact angles vary with time.

Load-bearing premise

The initial data must be sufficiently close to equilibrium so that the contact-point dynamics stay coupled to the surface evolution in the same controlled way for all future times.

What would settle it

An explicit solution starting close to equilibrium whose Sobolev norms of the interface or contact-point velocity become unbounded in finite time would disprove the global estimates.

Figures

Figures reproduced from arXiv: 2502.19286 by \'Angel Castro, Edoardo Bocchi, Francisco Gancedo.

Figure 1
Figure 1. Figure 1: Configuration of the vessel. The stated configuration is for a fluid filtered inside a vessel with a smooth boundary. The contact points where the fluid, the vessel, and the dry region meet occur along the vertical lateral walls, as shown in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A possible equilibrium configuration. together with non-penetration condition on the vessel (1.10) u · n = 0 on Γw(t), with n the outer normal vector to the vessel. We appointed this physical scenario as the Muskat problem [77] with contact points. Remarkably, it is mathematically analogous to the evolution of a fluid in a Hele-Shaw cell [84]. In it, two parallel plates are close enough together so that th… view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the different shapes of hs with zero mean in I: concave when JγK < 0, convex when JγK > 0 and flat when JγK = 0. 2.2. Fixed-boundary formulation. As usually done in free-boundary problems, we reformulate the problem in a fixed framework. Our choice for the reference domain in which we recast the problem is the stationary domain. Stationary solutions to (2.1)-(2.3) are couples (hs, ϕs), with ϕs ∈ R… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proves global-in-time a priori estimates for solutions to the 2D one-phase Muskat problem (Darcy's law) with contact points on vertical walls, initially close to equilibrium. The estimates are obtained in non-weighted L²-based Sobolev spaces without restrictions on contact angles by exploiting the Neumann problem solved by the velocity potential.

Significance. If the estimates hold, the result extends the contact-point analysis of Guo-Tice (Stokes 2018, Navier-Stokes 2024) to the more singular Muskat/Darcy setting and shows that closeness to equilibrium suffices to close global bounds in the absence of weights or angle restrictions. This would be a technically substantive contribution to free-boundary problems with moving contact lines.

major comments (2)
  1. [Main a priori estimate (energy identity and contact-point boundary terms)] The central claim that non-weighted H^s estimates close globally rests on controlling the contact-point traces and boundary integrals arising from the velocity potential. The skeptic correctly identifies that, under Darcy's law, the pressure gradient is recovered directly from the potential (no higher-order regularization), so any angle-dependent singularity at the moving contact point must be absorbed by the initial-closeness assumption. The manuscript must exhibit the precise cancellation or absorption mechanism (e.g., in the energy identity obtained after integration by parts against the Neumann solution) that works uniformly for angles away from π/2; without an explicit estimate of these terms, the non-weighted closure cannot be verified.
  2. [Section deriving the a priori bounds from the Neumann problem] The abstract states that the analysis is carried out 'without imposing restrictions on the contact angles,' yet the only mechanism cited for controlling the coupling is initial closeness to equilibrium. It is therefore necessary to show that the angle-dependent factors appearing in the trace theorems or in the commutator estimates at the contact points remain bounded by a constant independent of the angle once the solution stays sufficiently close to the flat equilibrium; otherwise the global bound may still require an implicit angle restriction.
minor comments (1)
  1. [Introduction / Setup] Notation for the contact-point velocity and the angle should be introduced with a diagram or explicit formulas in the setup section to avoid ambiguity when the angle deviates from π/2.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments, which help clarify the presentation of the contact-point estimates. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Main a priori estimate (energy identity and contact-point boundary terms)] The central claim that non-weighted H^s estimates close globally rests on controlling the contact-point traces and boundary integrals arising from the velocity potential. The skeptic correctly identifies that, under Darcy's law, the pressure gradient is recovered directly from the potential (no higher-order regularization), so any angle-dependent singularity at the moving contact point must be absorbed by the initial-closeness assumption. The manuscript must exhibit the precise cancellation or absorption mechanism (e.g., in the energy identity obtained after integration by parts against the Neumann solution) that works uniformly for angles away from π/2; without an explicit estimate of these terms, the non-weighted closure cannot be verified.

    Authors: We agree that the absorption of the contact-point boundary terms needs to be exhibited more explicitly. In the energy identity obtained by testing the Neumann problem for the velocity potential against itself (Section 3), integration by parts produces boundary integrals at the contact points involving the tangential derivative of the potential. These terms are controlled by the smallness of the interface displacement in the non-weighted Sobolev norm together with the kinematic condition at the contact point; the initial closeness ensures that the geometry remains a small perturbation of the flat state, allowing the singular factors to be absorbed into the dissipation without additional angle restrictions. To address the concern, we will insert a new lemma (Lemma 4.3 in the revision) that isolates and estimates these boundary integrals explicitly. revision: yes

  2. Referee: [Section deriving the a priori bounds from the Neumann problem] The abstract states that the analysis is carried out 'without imposing restrictions on the contact angles,' yet the only mechanism cited for controlling the coupling is initial closeness to equilibrium. It is therefore necessary to show that the angle-dependent factors appearing in the trace theorems or in the commutator estimates at the contact points remain bounded by a constant independent of the angle once the solution stays sufficiently close to the flat equilibrium; otherwise the global bound may still require an implicit angle restriction.

    Authors: The manuscript claims the absence of angle restrictions in the sense that no a-priori condition such as 'angle close to π/2' is imposed; for any fixed angle the smallness threshold on the initial data (which may depend on that angle) closes the estimates. The angle-dependent constants arising in the trace and commutator estimates at the contact points are therefore fixed once the angle is chosen and do not need to be uniform in the angle. We will add a clarifying paragraph in the introduction and a remark after the main theorem stating the dependence of the smallness constant on the contact angle. revision: partial

Circularity Check

0 steps flagged

No circularity: direct a priori estimate proof independent of its inputs

full rationale

The paper derives global-in-time a priori estimates for the one-phase Muskat problem via energy methods on the velocity potential's Neumann problem in non-weighted Sobolev spaces. No fitted parameters are renamed as predictions, no self-definitional loops appear in the contact-point coupling, and the cited Stokes/NS works are by unrelated authors (Guo-Tice) without load-bearing self-citation chains or imported uniqueness theorems. The initial closeness assumption is an explicit hypothesis required for closure, not a constructed output. The derivation is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard elliptic regularity for the Neumann problem and Sobolev embedding theorems; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard elliptic regularity and Sobolev embedding for the Neumann problem for the velocity potential
    Invoked to close the a priori estimates in non-weighted spaces

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discussion (0)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Global well-posedness of the one-phase Muskat problem with surface tension

    math.AP 2026-04 unverdicted novelty 8.0

    Global existence, uniqueness, and asymptotic decay to zero are shown for the one-phase Muskat problem with surface tension under small initial data in H^s for s > d/2 + 1.

  2. Global well-posedness of the one-phase Muskat problem with surface tension

    math.AP 2026-04 unverdicted novelty 8.0

    For initial free boundaries small in H^s with s > d/2 + 1, the one-phase Muskat problem with surface tension has a unique global strong solution that converges to the flat state in Lipschitz norm.

Reference graph

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