Failure of the Weak Sard property without Anomalous Dissipation

Umberto Pappalettera

Failure of the Weak Sard property without Anomalous Dissipation

T0 review · 0 major / 5 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read For every Hölder exponent below 1 there is a compactly supported autonomous 2D flow that fails the weak Sard property yet produces no anomalous dissipation.

desk verdict Clean counterexample: autonomous C^α fields can fail weak Sard without anomalous dissipation, refuting the BBDLM26 conjecture. read the letter →

arxiv 2607.27044 v2 pith:U6NMSDEH submitted 2026-07-29 math.AP

Umberto Pappalettera This is my paper

classification math.AP MSC 35Q3535B6576F25

keywords anomalousdissipationweakSardpropertyadvection-diffusionequationautonomousdivergence-freefieldsHamiltonianflowspassivescalarsmeasure-preservingdiffeomorphisms

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 constructs, for every smoothness exponent α between 0 and 1, a compactly supported divergence-free velocity field in the plane that is only C^α. The field fails the weak Sard property—a measure-theoretic condition on the critical set of its Hamiltonian that is already known to be necessary for anomalous energy loss—yet the associated advection-diffusion equation still sends the dissipated energy to zero as diffusivity vanishes. The construction repeatedly folds an annular region by measure-preserving maps that stretch and contract nested rectangles, yielding a limiting Hamiltonian whose critical set is large enough to break weak Sard, while intermediate smooth fields approximate the limit fast enough that diffusion cannot lock onto a positive energy fraction. The result disproves a conjecture that failure of a related Sard-type condition should be equivalent to anomalous dissipation. A reader who follows passive-scalar and 2D Hamiltonian transport cares because the example cleanly separates two phenomena that had been expected to travel together.

What carries the argument

A quantitative approximation criterion comparing viscous solutions for the limiting Hamiltonian H against inviscid transport for smooth approximants H_q: if ‖H−H_q‖_∞/κ_q → 0 and κ_q times the space-time integral of |DΦ_q|² vanishes, anomalous dissipation is ruled out. The H_q are built by composing measure-preserving rectangular quarter-turn maps on nested affine bulks inside a tubular neighborhood of a closed curve, so the flow in those coordinates is an explicit translation.

What would settle it

Along the paper’s explicit sequence κ_q ~ exp(−q/c) for the constructed limiting field, evaluate or rigorously lower-bound κ ∫ ‖∇θ^κ‖² for a fixed smooth compactly supported initial datum; a strictly positive limsup would falsify the central claim.

Watch

Extended reading notes

Core claim

For every α in (0,1) there exists a compactly supported autonomous divergence-free vector field u in C^α_c(R²,R²) whose Hamiltonian fails the weak Sard property, yet for every square-integrable initial datum the unique parabolic solution of the advection-diffusion equation satisfies lim_{κ↓0} κ ∫₀¹ ‖∇θ^κ_t‖²_{L²} dt = 0.

Load-bearing premise

The no-dissipation half of the theorem rests on choosing the stretching ratios and cutoff scales so that one sequence of smooth Hamiltonians both approximates the rough limit faster than diffusivity and keeps flow deformation from blowing up too fast.

Editorial extensions

If this is right

Failure of the weak Sard property is not sufficient for anomalous dissipation of passive scalars driven by autonomous 2D flows.
The conjecture equating failure of the relaxed Sard property with anomalous dissipation is false after periodic extension of the example.
Autonomous planar examples with non-unique inviscid transport can still be dissipation-regular under vanishing diffusion.
Future sufficient criteria for anomalous dissipation in this class must track quantitative stretching rates, not only the measure of the critical set.

Reading between the lines

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

The same nested-rectangle family appears tunable: slowing the approximation of H_q relative to κ should push the construction across a threshold into anomalous dissipation, giving a single-parameter bridge between the two regimes.
Periodic extension immediately places the counterexample on the torus, so the separation persists in the geometry most often used for idealised turbulence models.
Any successful criterion for anomalous dissipation in autonomous 2D flows will likely need a uniform positive lower bound on Lagrangian variance in the vanishing-noise limit, beyond mere failure of weak Sard.

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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

This paper gives an explicit autonomous divergence-free u in C^α_c(R^{2}) that fails weak Sard yet has vanishing anomalous dissipation for every L^{2} initial datum. That directly kills Conjecture 1.3 of Bagnara–Boutros–De Lellis–Mayboroda (up to the usual periodic extension). The two halves are independent and both check out.

What is new is the iterative rectangular quarter-turn construction. Start with a measure-preserving tubular parametrization of an annulus that is affine on a rectangle, then compose maps that contract width and expand height by a large factor V on a nested positive-measure “affine bulk” P_q while staying identity near the boundaries. The limiting Hamiltonian is C^{1,α}, the critical set contains Ψ(P_∞) of positive measure inside E*, and the push-forward is absolutely continuous and non-trivial, so weak Sard fails by direct computation. At the same time the same parameters keep ||H-H_q||_∞ decaying geometrically in (H/V)^q and the flow deformation of the smooth approximants only polynomially-in-q times V^{2q}. Choosing q ~ c log(1/κ) with c in a nonempty open interval makes both hypotheses of the comparison criterion (Prop. 4.3) hold, so dissipation vanishes.

The math is self-contained and carefully parameterized: the C^{1,α} Cauchy estimate, the L^∞ comparison lemma, the explicit flow formula Φ_q(t,Ψ_q(s,h))=Ψ_q(s+t,h), and the energy estimates are all written with explicit constants and compatible regimes for every α∈(0,1). Citations sit where they should (ABC, JS24, BBDLM26, DGG26). No circularity; the two conclusions are verified separately.

Soft spots are minor and mostly presentational. The construction is intricate (many nested rectangles, cut-offs, and free parameters V,W,H,δ_q), so a reader has to track the affine generations carefully, and there is no code. The setting is strictly 2-D autonomous, which is exactly the regime of the conjecture being disproved, so that is not a defect. The comparison criterion itself is natural and the parameter window is nonempty.

This is for people working on passive scalars, weak Sard, and necessary conditions for anomalous dissipation. It deserves a serious referee and should be engaged with; I would bring it to reading group and expect to cite the counter-example.

Referee Report

0 major / 5 minor

Summary. The paper constructs, for every α∈(0,1), a compactly supported autonomous divergence-free velocity field u∈C^α_c(R²,R²) whose Hamiltonian fails the weak Sard property, yet for which the associated advection-diffusion equation exhibits no anomalous dissipation of L² norm (Theorem 1.1). The construction proceeds by iteratively composing measure-preserving rectangular quarter-turn maps on a tubular neighborhood of a closed curve, producing a sequence of smooth Hamiltonians H_q converging in C^{1,α} to a limit H. Failure of weak Sard is read off from a positive-measure nested affine bulk P_∞ whose image under the limiting parametrization Ψ lies in S∩E* and pushes forward to a nontrivial absolutely continuous measure on the range. Absence of anomalous dissipation is obtained from a quantitative approximation criterion (Proposition 4.3) comparing the viscous solution for H to inviscid solutions for the smooth approximants H_q, once stretching parameters are chosen so that both ||H-H_q||_∞/κ_q o0 and κ_q∫∫|DΦ_q|² o0.

Significance. The result cleanly separates two properties that recent work had suggested might be equivalent for autonomous planar fields: failure of (weak/relaxed) Sard and anomalous dissipation. It thereby disproves Conjecture 1.3 of Bagnara-Boutros-De Lellis-Mayboroda. The argument is fully constructive and self-contained, with explicit diffeomorphisms, an explicit flow in adapted coordinates, and a usable comparison criterion (Proposition 4.3) that may be of independent interest. Parameter compatibility for every α∈(0,1) is checked carefully. This is a solid, definitive counterexample paper in the active area of passive-scalar anomalous dissipation.

minor comments (5)

[Lemma 2.1] Lemma 2.1 proof: typo "diffeomorphsims". Several other minor typos appear (e.g., spacing artifacts in the title block, "Dissip A TION").
[Section 2.4 / Corollary 4.4] The admissible range for (V,W,H,ε₁,ε₂,p,q₀) is scattered across (2.16), (2.23), Proposition 3.2 and Corollary 4.4. A short dedicated remark collecting one explicit admissible tuple (or the full list of inequalities) would help the reader verify compatibility at a glance.
[Lemma 3.3] In Lemma 3.3 the matrix M_q is defined with columns (∂_s Ψ_q, ∂_h Ψ_q), so det M_q ≡ -1, while (3.3) records det(∂_h,∂_s)=1. The signs are consistent but easy to misread; a one-line clarification would prevent confusion.
[Section 2] Figure 1 and Figure 3 are helpful; adding a brief caption note that shaded subrectangles in Figure 3 are exactly the affine bulk pieces belonging to P_δ would make the link to Definition 2.3 immediate.
[Proposition 5.1] The push-forward identity (5.1) is correct; it may be worth stressing explicitly that the full circle Ψ(T imes{h}) lies in the level set (not only the Cantor slice U^∞_x imes{h}), which is what guarantees the connected component is in E*.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: independent constructive verification of both halves of Theorem 1.1

full rationale

The paper is a pure existence construction. It builds an explicit sequence of measure-preserving diffeomorphisms Ψ_q (via rectangular quarter-turns β_q with parameters V,W,H,δ_q satisfying (2.16) and (2.23)), defines Hamiltonians H_q by (3.1), and passes to the C^{1,α} limit H (Proposition 3.2). Failure of weak Sard is then verified by direct computation: the nested affine bulk P_∞ has positive measure, Ψ(P_∞)⊂S∩E*, and H_#(1_{Ψ(P_∞)} L²)=L¹(U^∞_x)1_{U^∞_y} dh with both factors positive (Proposition 5.1, Corollary 5.2). Absence of anomalous dissipation is verified independently via the quantitative criterion Proposition 4.3: the same parameters yield ∥H−H_q∥_∞/κ_q o0 and κ_q∫∫|DΦ_q|² o0 (Corollary 4.4, using Lemmas 3.1, 3.3, 4.1, 4.2). Neither property is assumed in the definition of the other; parameter inequalities are chosen so both hold simultaneously, which is ordinary constructive bookkeeping, not a definitional reduction. Citations (ABC14, JS24, BBDLM26, EZs19, Pap25) supply background definitions or analogous techniques and are not load-bearing uniqueness claims that force the conclusion. No fitted inputs, self-definitional loops, or renamed empirical patterns appear.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The result rests on standard measure-theoretic and PDE facts (unique parabolic solutions, energy balance, area-preserving flows) plus the explicit iterative construction. The only free choices are the stretching parameters needed to close the C^{1,α} and deformation estimates; they are not fitted to data but chosen once to satisfy a finite list of inequalities.

free parameters (2)

V,W,H (stretching ratios) = any integers ≥2 obeying (2.16)
Positive integers/rationals satisfying W < V H < 1, H < W and 2 H V^α < W^{1+α}; chosen once so that both C^{1,α} Cauchy estimates and the flow-deformation bound close.
δ_q = (q+q_0)^{-p}, p>2, q_0≫1 = p>2, q_0 large
Decay rate of the transition-layer width in the quarter-turn maps; must be summable enough for igcap P_q to retain positive measure and for the C^k bounds to remain controllable.

assumptions (4)

domain assumption Unique parabolic solutions of advection-diffusion exist in C([0,1];L²)igcap L²([0,1];H¹) for divergence-free u∈C^α (BCC24)
Invoked to define the solution operator S^κ_H and the dissipation integral (1.2).
domain assumption Failure of weak Sard is necessary for anomalous dissipation of autonomous planar fields (JS24, Remark 1.3)
Used only for motivation; the paper proves the converse fails.
standard math Energy balance and integration-by-parts identities for the difference of two advection-diffusion solutions
Lemmas 4.1-4.2; classical.
standard math Tubular-neighborhood theorem and existence of smooth measure-preserving parametrizations of an annular region
Section 2.1; standard differential geometry.

invented entities (1)

Affine bulk / affine generation P_q of rectangular quarter-turn maps
purpose: Positive-measure nested sets on which each eta_q acts by pure anisotropic stretch, producing the critical set \Psi(P_∞) while keeping deformation bounds.
Defined in Definition 2.3 and (2.20); purely constructive, no independent physical existence claimed.

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Cite this review

Pith. "Pith review of Failure of the Weak Sard property without Anomalous Dissipation." pith.science (2026). https://pith.science/paper/U6NMSDEH

@misc{pith2026260727044,
  author       = {Pith},
  title        = {Pith review of: Failure of the Weak Sard property without Anomalous Dissipation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6NMSDEH}},
  note         = {Machine review of arXiv:2607.27044}
}

abstract

For every $\alpha\in(0,1)$ we construct an autonomous, divergence-free vector field $u \in C^\alpha_c(\mathbb{R}^2,\mathbb{R}^2)$ that does not have the weak Sard property and, nonetheless, does not induce anomalous dissipation of $L^2(\mathbb{R}^2)$ norm for solutions to the associated advection-diffusion equation. This disproves a conjecture proposed by Bagnara, Boutros, De Lellis and Mayboroda in \cite{BaBoDeMa26}.

Figures

Figures reproduced from arXiv: 2607.27044 by the authors.

**Figure 1.** On the highlighted constant-speed straight segment, κ(s) ≡ 0 and h(s, r) = r. Since Ψ0 is affine on R0, the image Ψ0(R0) is a rectangle compactly contained in A. For s ∈ T let τ (s) denote the tangent versor to the curve γ at the point γ(s) and choose a normal versor n(s) such that the basis of R 2 (n(s), τ (s)) is positively oriented. Define the curvature κ(s) by n ′ (s) =: −κ(s)τ (s), [PITH_FULL_IMAGE:figures/ful… view at source ↗

**Figure 2.** The map Tδ is an exact 90◦ rotation inside the small square (1 − C⋆δ)Q, the identity outside of the large square (1 − c⋆δ)Q, and smooth in between. Step 2. Next, we want to parametrize points (x, y) ∈ ∂Dδ in polar coordinates. For every angle ϕ ∈ [0, 2π) between the x-axis and the vector (x, y), we have that (x, y) =: ρδ(ϕ)(cos(ϕ),sin(ϕ)) ∈ ∂Dδ if and only if ρδ(ϕ) Nδ [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Action of Bδ on a representative rectangle Ri,j . Colored subrectangles belong to the set Pδ, on which Bδ is affine, contracts width by a factor V −1 and expands height by a factor V . Shades of colors indicate the orientation of the rectangles at each step. Next, we want to determine some properties of the set Pδ. Denote C⋆ the constant from Lemma 2.2 and introduce λδ := 1 − C⋆δ 1/2 . Inside every column Ri,j,k ta… view at source ↗

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