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arxiv: 2606.22384 · v1 · pith:BGD6AZVQnew · submitted 2026-06-21 · 🧮 math.AP

Global Well-Posedness of sessile drop problem: 2D Navier-Stokes Flow

Xiaoding Yang This is my paper

Pith reviewed 2026-06-26 10:22 UTC · model grok-4.3

classification 🧮 math.AP
keywords sessile dropNavier-Stokesglobal well-posednesssurface tensionmoving coordinatescontact lineNavier-slip
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The pith

Small perturbations of two-dimensional sessile droplet equilibria admit unique global solutions for the Navier-Stokes equations with surface tension.

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

The paper proves that small perturbations around equilibrium configurations of a two-dimensional sessile drop evolve for all future times under the incompressible Navier-Stokes equations equipped with surface tension, Navier-slip boundary conditions, and a dynamic contact-point law. The main technical step is to eliminate the horizontal translational invariance of the equilibria by adopting a moving polar coordinate frame whose origin is fixed through an orthogonality condition. Local existence follows from a Galerkin construction of pressureless weak solutions, pressure recovery, higher-order estimates, and a contraction mapping argument; these local solutions are then continued globally by invoking energy-dissipation estimates established in earlier work, which also furnish exponential decay to equilibrium.

Core claim

Reformulating the free-boundary Navier-Stokes system in a moving polar coordinate system chosen so that an orthogonality condition holds removes the translational degeneracy, allowing a Galerkin-based local well-posedness theory that combines with prior global energy estimates to produce unique global solutions decaying exponentially to the sessile equilibrium.

What carries the argument

Moving polar coordinate system fixed by an orthogonality condition that removes the horizontal translational degeneracy of the equilibrium manifold.

If this is right

  • Unique global-in-time solutions exist for all sufficiently small initial perturbations.
  • The solutions converge exponentially fast to the equilibrium in suitable function spaces.
  • The local well-posedness construction via Galerkin approximation and contraction extends the local theory to the global regime when combined with energy decay.
  • The result holds specifically under Navier-slip conditions and the dynamic contact-point law.

Where Pith is reading between the lines

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

  • The same moving-frame technique could be applied to three-dimensional sessile drops once corresponding energy estimates are available.
  • Other free-boundary fluid problems with translational symmetry might benefit from analogous coordinate adjustments to restore well-posedness.
  • Direct numerical simulations of the system could verify the predicted exponential decay rates for small perturbations.

Load-bearing premise

The global energy-dissipation estimates obtained in the authors' previous work continue to hold for the perturbed solutions constructed in the moving polar coordinate frame.

What would settle it

An explicit construction of a small initial perturbation whose solution either ceases to exist after finite time or violates the exponential decay bound in the moving frame would disprove the global well-posedness claim.

Figures

Figures reproduced from arXiv: 2606.22384 by Xiaoding Yang.

Figure 1
Figure 1. Figure 1: A droplet. References 60 1. Introduction 1.1. Formulation and Origins of the Problem. Consider a two-dimensional droplet of viscous incom￾pressible fluid evolving above a one-dimensional flat surface. The interface between this droplet and the vapor may not be a graph of x in Cartesian coordinates. See [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

We prove the global well-posedness of small perturbations of two-dimensional sessile droplet equilibria for the incompressible Navier--Stokes equations with surface tension, Navier-slip boundary conditions, and a dynamic contact-point law. The main difficulty is the construction of solutions in the presence of the horizontal translational degeneracy of the equilibrium manifold. To remove this degeneracy, we work in a moving polar coordinate system determined by an orthogonality condition. We then establish local well-posedness through a Galerkin construction of pressureless weak solutions, recovery of the pressure, higher-order estimates, and a contraction argument. Combining this local theory with the global energy--dissipation estimates obtained in our previous work yields a unique global solution and the corresponding exponential decay estimate.

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

1 major / 1 minor

Summary. The manuscript proves global well-posedness and exponential decay for small perturbations of 2D sessile droplet equilibria in the incompressible Navier-Stokes system with surface tension, Navier-slip boundary conditions, and a dynamic contact-point law. Translational degeneracy of the equilibrium manifold is removed by working in a moving polar coordinate frame chosen via an orthogonality condition. Local well-posedness is obtained by a Galerkin construction of pressureless weak solutions, followed by pressure recovery, higher-order estimates, and a contraction argument; global existence then follows by combining the local theory with global energy-dissipation estimates from the authors' prior work.

Significance. If the result holds, it would constitute a substantial advance in the mathematical analysis of free-boundary incompressible flows with moving contact lines. The strategy of removing degeneracy via a moving frame and extending local solutions with a priori energy estimates is standard, yet its successful implementation here would resolve a central open question for this physically relevant 2D model. The manuscript would thereby supply the first global existence theorem with decay for small data in this setting.

major comments (1)
  1. [Abstract, final paragraph] Abstract, final paragraph: the global existence and decay statements rest on the claim that the global energy-dissipation estimates from the authors' previous work continue to hold for the solutions constructed in the time-dependent moving polar frame. Because the frame is time-dependent, the transformed velocity, height, and contact-point variables may generate commutator terms in the energy identity or modify the dissipation structure; the manuscript must explicitly derive or verify that these estimates remain valid without additional uncontrolled terms.
minor comments (1)
  1. The abstract refers to both 'sessile drop' and 'sessile droplet'; consistent terminology throughout the manuscript would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the applicability of the prior energy estimates. We address the point below.

read point-by-point responses

  1. Referee: [Abstract, final paragraph] Abstract, final paragraph: the global existence and decay statements rest on the claim that the global energy-dissipation estimates from the authors' previous work continue to hold for the solutions constructed in the time-dependent moving polar frame. Because the frame is time-dependent, the transformed velocity, height, and contact-point variables may generate commutator terms in the energy identity or modify the dissipation structure; the manuscript must explicitly derive or verify that these estimates remain valid without additional uncontrolled terms.

    Authors: We agree that the time-dependent moving polar frame requires an explicit check that the energy-dissipation identities from the previous work remain valid. In the revised version we will insert a dedicated subsection (immediately after the coordinate change is introduced) that recomputes the energy identity in the moving frame, identifies all commutator terms arising from the time-dependent transformation, and shows that these terms are either identically zero by the orthogonality condition or controlled by the smallness of the perturbation and absorbed into the dissipation. With this verification the global existence and decay statements follow directly from the local theory and the a-priori estimates. revision: yes

Circularity Check

0 steps flagged

No significant circularity; local construction is independent and prior estimates treated as external input

full rationale

The paper's chain is: (1) introduce moving polar frame via orthogonality to remove degeneracy, (2) prove local well-posedness by Galerkin + pressure recovery + contraction in that frame, (3) invoke global energy-dissipation estimates from prior work to upgrade to global existence/decay. The abstract explicitly presents the energy estimates as 'obtained in our previous work' and independent of the new local theory. No equation reduces to an input by definition, no fitted parameter is relabeled as prediction, and the self-citation is not load-bearing in the circular sense because the prior result is external, not derived from the present local construction or frame change. The derivation remains self-contained against the cited external estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard functional-analytic tools (Galerkin approximation, weak-solution theory for Navier-Stokes) and on energy-dissipation estimates from prior work by the same author; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Galerkin approximations converge to a pressureless weak solution that can be upgraded to a strong solution satisfying the dynamic contact law.
    Invoked in the local well-posedness step described in the abstract.
  • domain assumption The energy-dissipation estimates from the authors' previous work remain valid after the change to the moving polar coordinate frame.
    Central step that upgrades local existence to global existence and decay.

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Reference graph

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