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arxiv: 2607.00677 · v1 · pith:RWLVRMHWnew · submitted 2026-07-01 · 🧮 math.AP

Simultaneous Reconstruction of Multiple Unknowns in Stokes-Darcy System from Partial Boundary Data

Yu Jia , Huanzhao Ren , Qu Fenglong , Jiaqing Yang This is my paper

Pith reviewed 2026-07-02 09:48 UTC · model grok-4.3

classification 🧮 math.AP
keywords Stokes-Darcy systeminverse boundary value problemglobal uniquenessinterior transmission problempartial boundary dataviscosity coefficientembedded objectcoupled system
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The pith

All three unknowns in a Stokes-Darcy system are uniquely determined by localized boundary Cauchy data.

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

The paper proves that the viscosity coefficient, the fluid-porous interface, and an embedded solid object inside the free-flow region can be recovered simultaneously from measurements taken on only part of the outer boundary. It introduces a construction of an interior transmission problem whose solutions are made to carry strong singularities that separate the effects of each unknown. A reader would care because this removes the need to know any of the three quantities in advance when solving the inverse problem for the coupled system. The result is stated as a global uniqueness theorem rather than a conditional or local recovery.

Core claim

The paper establishes a global uniqueness theorem showing that the viscosity coefficient μ, the interface Γ, and the internal object D are all uniquely determined by the boundary measurements. A novel method based on the construction of an interior transmission problem is introduced, which amplifies the singularity of solutions to distinguish the three unknowns from the localized Cauchy data.

What carries the argument

An interior transmission problem constructed to amplify the singularity of solutions sufficiently to distinguish the viscosity coefficient, interface, and internal object from localized Cauchy data.

If this is right

  • The three unknowns can be recovered simultaneously rather than sequentially.
  • Only partial boundary measurements suffice for the global uniqueness result.
  • The construction applies directly to the coupled Stokes-Darcy model with an embedded object.
  • The amplified-singularity approach yields a theorem that holds without additional regularity assumptions on the unknowns.

Where Pith is reading between the lines

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

  • Numerical reconstruction schemes could be built that iterate over candidate interior transmission problems to match observed data.
  • The same singularity-amplification idea might transfer to other coupled interface problems such as fluid-structure interactions.
  • Synthetic experiments that deliberately vary the strength of the constructed singularities could test how robust the separation of unknowns remains under noise.

Load-bearing premise

The interior transmission problem can be constructed so that it amplifies the singularity of solutions sufficiently to distinguish the three unknowns from the localized Cauchy data.

What would settle it

Two different combinations of viscosity coefficient, interface location, and embedded object that produce identical localized Cauchy data on the boundary would disprove the uniqueness result.

Figures

Figures reproduced from arXiv: 2607.00677 by Huanzhao Ren, Jiaqing Yang, Qu Fenglong, Yu Jia.

Figure 1.1
Figure 1.1. Figure 1.1: Geometrical illustration. pressure ϕp : Ωp → R is described as follows:    −µ∆uf + ∇pf = 0 in Ωf \ D, divuf = 0 in Ωf \ D, uf = f on Γf , uf = 0 on ∂D −∇ · (K∇ϕp) = 0 in Ωp, ϕp = 0 on Γp, (1.1) with a general setting on the interface Γ: uf · n = − K∇ϕp · n on Γ, (1.2) −n · T(uf , pf ) · n =ϕp on Γ, (1.3) uf · τi + ατi · T(uf , pf ) · n =0 on Γ, (1.4) where µ is the viscosity constant a… view at source ↗
read the original abstract

This paper studies an inverse boundary value problem for a coupled Stokes-Darcy system modeling fluid-porous medium interaction, with an unknown solid object embedded in the free-flow region. We simultaneously recover the viscosity coefficient $\mu$, the interface $\Gamma$, and the internal object $D$ from localized boundary Cauchy data. A novel method based on the construction of an interior transmission problem is introduced, which can amplify the singularity of solutions. We establish a global uniqueness theorem, showing that all three unknowns are uniquely determined by the boundary measurements.

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

0 major / 0 minor

Summary. The paper studies an inverse boundary value problem for a coupled Stokes-Darcy system modeling fluid-porous medium interaction with an unknown solid object embedded in the free-flow region. It claims to simultaneously recover the viscosity coefficient μ, the interface Γ, and the internal object D from localized boundary Cauchy data by introducing a novel method based on the construction of an interior transmission problem that amplifies the singularity of solutions, and establishes a global uniqueness theorem showing that all three unknowns are uniquely determined by the boundary measurements.

Significance. If the result holds, this would constitute a notable contribution to inverse problems for coupled PDE systems, as the simultaneous recovery of a coefficient, an interface, and a subdomain from partial (localized) Cauchy data is technically demanding. The approach of using an interior transmission problem to generate solutions with controllable singularities offers a potentially reusable technique for other transmission or interface inverse problems in fluid mechanics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the significance of our result on simultaneous recovery of μ, Γ, and D in the Stokes-Darcy system from localized Cauchy data. The referee notes that the interior transmission problem construction is potentially reusable; we agree this is a key technical feature. No specific major comments were provided in the report, so we address the overall assessment below.

Circularity Check

0 steps flagged

No significant circularity; uniqueness theorem is self-contained PDE analysis

full rationale

The paper claims a global uniqueness result for the triple (μ, Γ, D) recovered from localized Cauchy data on a Stokes-Darcy system. The method relies on constructing an interior transmission problem to produce controllable singularities that distinguish the unknowns. No parameter fitting occurs, no predictions are made from fitted inputs, and no load-bearing steps reduce to self-citations or self-definitional loops. The derivation is a standard inverse-problem uniqueness argument via special solutions and density arguments; it does not rename known results or smuggle ansatzes via prior self-work. The central claim therefore stands as independent mathematical content rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are visible. The result rests on standard PDE well-posedness and transmission conditions for Stokes-Darcy systems.

axioms (1)
  • domain assumption The coupled Stokes-Darcy system admits well-posed solutions in appropriate Sobolev spaces under standard interface conditions.
    Implicit in any analysis of the forward problem for the inverse result.

pith-pipeline@v0.9.1-grok · 5613 in / 1046 out tokens · 18784 ms · 2026-07-02T09:48:35.122223+00:00 · methodology

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Reference graph

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