A Calder\'on type inverse problem for the active scalar equations with fractional dissipation

Li Li; Weinan Wang

REVIEW 2 major objections 3 minor 1 cited by

A Calder\'on type inverse problem for the active scalar equations with fractional dissipation

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Observing $\theta$ and $u$ in an arbitrarily small open set $W$ uniquely determines the nonlocal drift operator $R$ on the exterior $W^e$ for dissipative active scalar equations with fractional dissipation.

desk verdict Genuinely novel Calderón-type inverse problem for active scalar equations with fractional dissipation, but Proposition 4.3 is algebraically wrong, so Theorem 1.1 is not proven as written. read the letter →

arxiv 2412.03868 v2 pith:BS5VVXSK submitted 2024-12-05 math.AP

Li Li , Weinan Wang This is my paper · ORCID

classification math.AP MSC 35R3035R1135Q35

keywords activescalarequationsfractionaldissipationCalderóninverseproblemsecond-orderlinearizationLaplacianuniquecontinuationRungeapproximationnonlocaldriftoperator

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies an inverse problem for dissipative active scalar equations on the two-dimensional torus, where a scalar field $\theta$ is carried by a divergence-free velocity $u = R\theta$ generated from $\theta$ through a nonlocal operator $R$ of order $-1$. The authors claim that if two such operators $R_1$ and $R_2$ produce the same observations of both $\theta$ and $u$ in a fixed, arbitrarily small open set $W$, then $R_1$ and $R_2$ must agree on all smooth functions supported in the exterior $W^e$. In other words, a tiny interior window of data determines the nonlocal drift mechanism outside the window. The reason such a statement can be true is that the fractional dissipation $(-\Delta)^\alpha$ with $\alpha > 1/2$ gives the equation nonlocal unique continuation, so information from the observation region reaches the whole torus, while the quadratic nonlinearity lets the inverse problem be reduced to a linear fractional diffusion equation. If correct, this is the first Calderón-type uniqueness result for this class of fractional active scalar equations.

What carries the argument

The argument is carried by three linked mechanisms. First, second-order linearization: the mixed second difference quotient of the solution map, built from two small sources supported in $W$, converges to a linear equation whose source is $-Rw_1 \cdot \nabla w_2 - Rw_2 \cdot \nabla w_1$, and this is the step that turns recovery of a nonlinear operator into recovery of a term in a linear equation. Second, the unique continuation property of the fractional Laplacian on the torus, which states that a sufficiently regular function with $(-\Delta)^r u = u = 0$ in any nonempty open set vanishes everywhere; this is what forces agreement of the two nonlinear solutions to propagate from $W$ to the whole torus. Third, a Runge approximation property for the linear fractional diffusion equation, which says that solutions sourced in $W$ are dense in $L^2(0,T;L^2(T^2 \setminus W))$ when restricted to the exterior; this is what converts the final integral identities into statements about arbitrary test functions and produces the kernel-difference condition. The divergence-free structure $\mathrm{div}\,R\theta = 0$ is used throughout, in the uniqueness argument and in the integration by parts.

What would settle it

A decisive check is to compute the difference between $S_\epsilon$ as defined in Eq. (4.9) and its four-term decomposition in Eq. (4.10) for an explicit choice of $R$ (for instance the periodic Riesz transform) and two sources supported in $W$. If the remainder contains self-interaction terms of the form $R(w_1-w_2)\cdot\nabla(w_1-w_2)$ whose $L^2(0,T;H^{-\alpha})$ norm does not tend to zero as $\epsilon \to 0$, then Proposition 4.3 and the theorem's linearization step are not established.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for $T>0$ arbitrarily small, if the source-to-solution maps $L_{R_1}$ and $L_{R_2}$ agree for all sources $f$ supported in $W \times (0,T)$, where each map records the restriction of $\theta$ and $u$ to $W \times (0,T)$, then $R_1g|_{W^e} = R_2g|_{W^e}$ for every smooth $g$ supported in $W^e$. The discovery is that the nonlocal drift operator $R$, a Fourier-multiplier object rather than a local coefficient, can be recovered outside the observation region from interior data. The proof uses a second-order linearization: with two small sources $f_1$ and $f_2$ supported in $W$, the mixed second difference quotient of the solution map converges to the solution $v$ of $\partial_t v + R w_1 \cdot \nabla w_2 + R w_2 \cdot \nabla w_1 + (-\Delta)^\alpha v = 0$, where $w_1, w_2$ solve the linear fractional diffusion equation. Equality of the two maps forces this mixed term to agree for $R_1$ and $R_2$; the divergence-free property of $R$ permits an integration by parts, unique continuation of the fractional Laplacian globalizes equality from $W$ to the torus, and a Runge approximation property lets the resulting identities be tested against arbitrary functions in $W^e$, yielding the exterior equality of $R_1$ and $R_2$.

Load-bearing premise

The argument's load-bearing premise is that the nonlinear response to two small sources can be split cleanly into first-order and second-order pieces with no leftover cross-terms; if those cross-terms do not vanish, the equation that determines $R$ is not the one the proof analyzes.

Editorial extensions

If this is right

If the theorem is correct, measurements taken in any arbitrarily small open set, with no boundary access, determine the drift operator $R$ on the complement of that set.
For models such as the surface quasi-geostrophic equation, where $R$ is a Riesz-type operator, the result gives exterior uniqueness for the velocity law from local observations of the scalar and velocity.
Because $W$ can be chosen arbitrarily small, the uniqueness statement is insensitive to the size of the observation window.
The proof gives a template for nonlinear fractional inverse problems: second-order linearization combined with unique continuation can recover nonlocal terms rather than only local variable coefficients.
The exterior-only conclusion is a genuine limitation of the method, since the Runge approximation is available only outside $W$; determination on the whole torus is left open.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the authors do not pursue is higher-order linearization: third- or fourth-order mixed terms encode iterated compositions of $R$ and might allow recovery of more of the operator's structure, possibly on the whole torus.
The proof is qualitative, but a quantitative version of the Runge approximation could turn the uniqueness statement into a stability estimate, likely logarithmic or Hölder, for the difference $R_1-R_2$ in terms of the discrepancy between the two source-to-solution maps.
A numerical check of the second-order linearization on a known kernel would isolate whether the convergence of the mixed difference quotient is the only delicate step; this is a testable extension of the paper's analytic argument.
If the divergence-free condition were dropped, the integration-by-parts step that removes velocity terms would fail, so some form of velocity information appears essential; identifying the minimal measurements is a separate question.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is the first Calderón-style inverse problem I know of for active scalar equations with fractional dissipation, and the unknown is the nonlocal drift operator R rather than a local coefficient — a real novelty worth attention. Second, the central proof step, Proposition 4.3, is wrong as written; the algebra does not close, so Theorem 1.1 is not established in the submitted version.

The paper has real merits. The problem is well chosen (it includes SQG), the well-posedness and Runge/unique continuation machinery are quoted from the literature and look standard, and the final argument recovering R on W^e from the bilinear identity is clean. The authors are also honest about the limitation to the exterior.

The soft spot is load-bearing. In the proof of Prop 4.3, Eq. (4.10) claims S_epsilon is the sum of four terms. Direct expansion with A = θ_{ε(f1+f2)}, B = θ_{εf1}, C = θ_{εf2}, and N(U,V) = R(U)·∇V shows the sum contains extra terms −N(A,B) + N(B,B) + N(A,C) − 2N(B,C) compared with S_epsilon. In the ε→0 limit the asserted convergence of each S_epsilon,j to zero fails; the leftover is N(w1,w1) + N(w2,w2) − N(w1,w2) − N(w2,w1), which is nonzero for generic w1,w2. Since Prop 4.3 is the only bridge from the second-order linearization to v(1) = v(2) and the identity (R1−R2)w1·∇w2 + (R1−R2)w2·∇w1 = 0, the main theorem is unsupported.

My guess is this is a repairable slip rather than a dead end: a correct polarization identity should produce the desired bilinear expression, and the rest of the paper (Runge approximation, the recovery of R from the identity) seems fine. But as submitted, the proof does not go through.

Who is this for? People working on inverse problems for nonlocal or nonlinear PDEs will find the setup and the strategy worth reading, even if the central proof needs repair. It deserves a serious referee — send it to peer review with a clear request to fix the second-order linearization. If the repair works, it is a solid contribution.

Referee Report

2 major / 3 minor

Summary. The paper studies an inverse problem for the dissipative active scalar equation ∂tθ + Rθ·∇θ + (−∆)αθ = f on the torus T^2, with α ∈ (1/2,1) and with drift velocity u = Rθ given by a divergence-free Fourier multiplier of order −1 (for example, the SQG Riesz-vector case). The main result, Theorem 1.1, asserts that if the source-to-solution maps LR1 and LR2, which record both θ and u in an arbitrarily small observation window W × (0,T), agree for all sources supported in W, then R1g = R2g on the exterior We = T^2 \ W̄ for all g ∈ Cc∞(We). The proof combines forward well-posedness, first- and second-order linearizations, a unique continuation property of the fractional Laplacian, a Runge approximation property for the linear fractional heat equation, and a final convolution-kernel argument. The paper is clearly organized and the overall strategy is standard for the fractional Calderón program, but the key second-order linearization proposition is not proven correctly as written.

Significance. If the main theorem is correct, it is a meaningful extension of the fractional Calderón methodology to nonlinear active scalar equations, and it is unusual in that the recovered object is the nonlocal drift operator R rather than a local coefficient. The paper is well written, the well-posedness result is quoted from the SQG literature with a proof sketch, and the UCP/Runge ingredients are standard and used naturally. There are no fitted parameters and no circular assumptions. The obstruction is the proof of Proposition 4.3, which is the sole bridge from the second-order linearization to the determining identity used in Section 6; as submitted, that proof contains a false algebraic decomposition and an unjustified convergence claim. The central idea is defensible, but the manuscript needs a corrected proof of Proposition 4.3 before the main theorem can be accepted.

major comments (2)

[§4, Proposition 4.3, Eq. (4.9) and the displayed decomposition after it] The identity Sε = Sε,1 + Sε,2 + Sε,3 + Sε,4 is algebraically false. Write A = θε(f1+f2), B = θεf1, C = θεf2, and N(U,V) = R(U)·∇V. The sum of the four numerators is [N(A,A) − N(A,B) + N(B,B) + N(A,C) − N(C,C) − 2N(B,C)]/ε², whereas the numerator of Sε is [N(A,A) − N(B,B) − N(C,C)]/ε². The difference does not vanish. Using the first-order asymptotics A ≈ ε(w1+w2), B ≈ εw1, C ≈ εw2, the residual is N(w1,w1) + N(w2,w2) − N(w1,w2) − N(w2,w1), which is nonzero for generic w1,w2. In particular, Sε,1 tends to N(w2,w2) − N(w2,w1) and Sε,2 tends to N(w1,w1) − N(w1,w2), so the assertion that each Sε,j converges to 0 in L2(0,T;H−α) is false. Since Proposition 4.3 is the only step that produces v(1) = v(2) and hence the identity (R1−R2)w1·∇w2 + (R1−R2)w2·∇w1 = 0 used in Section 6, Theorem 1.1 is not established by the submitted proof. The proposition is likely repairable by a standard two-term expansion θεf = εw + ε²q + o(ε²), but a correct proof must be supplied.
[§4, proof of Proposition 4.3] Even apart from the algebraic error, the asserted convergence Sε,j → 0 in L2(0,T;H−α) is not justified by the displayed embedding argument. For example, Sε,1 is the product of R(θε(f1+f2)−θεf1)/ε and ∇(θε(f1+f2)−θεf1)/ε; by Proposition 4.2 both factors converge in H2α−1, not one in Hα. The embedding chain H2α−1 ↪ L1/(1−α) ↪ L1/α places each factor in L1/α, but the multiplier statement for L1/α does not control the product of two such factors. A different product estimate is needed even for a corrected decomposition.

minor comments (3)

[§5, proof of Proposition 5.2] After deriving v = 0 in W × (0,T), the text says 'so (−∆)αv = 0 in W × (0,T)'; this uses the fact that ∂tv = 0 in W as well, which follows from v = 0 but is not stated explicitly.
[§6, proof of Theorem 1.1] The passage from the time-integrated identity to the pointwise identity 'in particular' would be clearer if it explicitly used arbitrary time cutoffs χ(t) to remove the time integral.
[§4 and §6, notation] The notation W e = T2 \ W̄ is introduced in the introduction, but later the text sometimes writes T2 \ W; standardizing this notation would avoid ambiguity about whether the boundary of W is included.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is derived from external unique continuation and Runge approximation results, and no fitted input is renamed as a prediction.

full rationale

The paper's derivation chain is not circular. Theorem 1.1 aims to recover the operator R on the exterior set W^e from equality of two source-to-solution maps L_{R1} and L_{R2}; the proof does not fit any parameter to the data that it later 'predicts.' The assumptions (1.3)-(1.5) are stated at the outset and are not chosen to force the conclusion. The linearization arguments in Propositions 4.2 and 4.3 expand the nonlinear solutions in powers of epsilon and produce the identity (R1-R2)w1·∇w2 + (R1-R2)w2·∇w1 = 0; the recovery of R then follows from the unique continuation property and the Runge approximation property for the fractional Laplacian, which are cited from external works [9], [5], and [20], not from the authors' own prior results. The self-citations in the paper ([16], [17]) and the co-authored reference [11] appear only as contextual examples of related inverse problems or active scalar models; none of them supplies a load-bearing premise of Theorem 1.1. Even if the skeptic's objection to Proposition 4.3 is correct, that objection alleges an algebraic error in an intermediate expansion, not that a conclusion has been assumed as an input; a false step is a correctness defect, not circularity. There are no fitted inputs renamed as predictions, no uniqueness theorem imported solely from the authors, and no ansatz smuggled in via citation. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities: the theorem is a conditional uniqueness statement with structural assumptions on the kernel K. The main external inputs are prior unique continuation, well-posedness, and approximation results, none of which encode the conclusion.

assumptions (4)

standard math Unique continuation property for the fractional Laplacian on T2 (entanglement principle).
Invoked in Prop 5.1 and used twice in the main proof; quoted from [5, Theorem 1.8].
standard math Well-posedness of the nonlinear active scalar equation for alpha in (1/2,1), including Lq and H^s estimates.
Proposition 3.1 is taken from [18, Thm 3.5, 3.7] and sketched in the appendix.
standard math Well-posedness and regularity for the linear fractional diffusion equation.
Proposition 4.1 is stated without proof as a standard Galerkin result; used for linearization estimates.
standard math Sobolev embedding and multiplier estimates used in the appendix.
Standard results, with the product lemma quoted from [18, Lemma A.4].

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Cite this review

Pith. "Pith review of A Calder\'on type inverse problem for the active scalar equations with fractional dissipation." pith.science (2026). https://pith.science/paper/BS5VVXSK

@misc{pith2026241203868,
  author       = {Pith},
  title        = {Pith review of: A Calder\'on type inverse problem for the active scalar equations with fractional dissipation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BS5VVXSK}},
  note         = {Machine review of arXiv:2412.03868}
}

read the original abstract

In this paper, we are interested in an inverse problem for the active scalar equations with fractional dissipation on the torus. We perform a second order linearization to relate our model to the linear fractional diffusion equation. Our approach to solving the inverse problem relies on nonlocal phenomena such as the unique continuation property of the fractional Laplacian and its associated Runge approximation property. A remarkable feature of our model is that the divergence-free structure in the nonlinear term plays an important role in both forward and inverse problems.

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Forward citations

Cited by 1 Pith paper

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

Entanglement principle for the fractional Laplacian with applications to inverse problems
math.AP 2024-12 conditional novelty 7.0 of 10

A unique-continuation principle for sums of fractional Laplacians is proved on Euclidean space and applied to recover anisotropic coefficients and potentials in fractional polyharmonic equations.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

On instability properties of the fractional Calder´ on problem

Hendrik Baers, Giovanni Covi, and Angkana R¨ uland. On instability properties of the fractional Calder´ on problem. arXiv preprint arXiv:2405.08381 , 2024

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work page 2012

[3] [3]

The higher order fractional Calder´ on problem for linear local operators: Uniqueness

Giovanni Covi, Keijo M¨ onkk¨ onen, Jesse Railo, and Gunther Uhlmann. The higher order fractional Calder´ on problem for linear local operators: Uniqueness. Advances in Mathematics , 399:108246, 2022

work page 2022

[4] [4]

Partial differential equations , volume 19

Lawrence C Evans. Partial differential equations , volume 19. American mathematical society, 1998

work page 1998

[5] [5]

Calder´ on problem for fractional Schr¨ odinger operators on closed Riemannian manifolds

Ali Feizmohammadi, Katya Krupchyk, and Gunther Uhlmann. Calder´ on problem for fractional Schr¨ odinger operators on closed Riemannian manifolds. arXiv preprint arXiv:2407.16866 , 2024

arXiv 2024

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An inverse problem for a semi-linear elliptic equation in riemannian geometries

Ali Feizmohammadi and Lauri Oksanen. An inverse problem for a semi-linear elliptic equation in riemannian geometries. Journal of Differential Equations , 269(6):4683–4719, 2020

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[7] [7]

Ill/well-posedness of non-diffusive active scalar equations with physical applications

Susan Friedlander, Anthony Suen, and Fei Wang. Ill/well-posedness of non-diffusive active scalar equations with physical applications. Journal of Differential Equations , 411:880–902, 2024

work page 2024

[8] [8]

Uniqueness and reconstruction for the fractional Calder´ on problem with a single measurement.Journal of Functional Analysis , page 108505, 2020

Tuhin Ghosh, Angkana R¨ uland, Mikko Salo, and Gunther Uhlmann. Uniqueness and reconstruction for the fractional Calder´ on problem with a single measurement.Journal of Functional Analysis , page 108505, 2020

work page 2020