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arxiv: 2604.13912 · v1 · submitted 2026-04-15 · 🧮 math.AP

Scalar anomalous dissipation and optimal regularity via iterated homogenization

Jan Burczak , L\'aszl\'o Sz\'ekelyhidi , Jr. , Bian Wu This is my paper

Pith reviewed 2026-05-10 12:21 UTC · model grok-4.3

classification 🧮 math.AP
keywords anomalous dissipationadvection-diffusion equationdivergence-free vector fieldsHölder regularityiterated homogenizationscalar mixingenergy dissipationtransport equations
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The pith

Divergence-free vector fields with Hölder regularity below 1/3 drive anomalous dissipation in advection-diffusion for arbitrary initial data while keeping the scalar in a complementary Hölder class.

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

The paper constructs divergence-free vector fields that are only Hölder continuous with any exponent strictly less than 1/3 in space and time. These fields are paired with a sequence of diffusivities shrinking to zero. For any initial density from a low-regularity class, the corresponding solution to the advection-diffusion equation loses energy at a positive rate that does not vanish along the sequence. The solution density remains uniformly bounded in a Hölder space whose regularity index satisfies the relation that twice this index plus the velocity index is less than one. This shows that the critical threshold for anomalous dissipation can be reached through iterated homogenization and that the dissipation anomaly can be arranged to be independent of time.

Core claim

For any β₀ < 1/3 the authors produce a divergence-free velocity field in the Hölder class C^{β₀} in space and time together with a sequence of positive diffusivities κ_q tending to zero. The classical solutions ρ_q of the advection-diffusion equation then exhibit anomalous dissipation along this sequence for every initial datum taken from a sufficiently regular but otherwise arbitrary low-regularity class. The same solutions remain bounded in the space C^0_t C^{α₀}_x whenever β₀ + 2α₀ < 1. The construction proceeds by iterated homogenization, which supplies the required mixing rates at the stated regularity while preserving the divergence-free condition and the uniform scalar bound.

What carries the argument

Iterated homogenization, which successively refines the velocity field at smaller and smaller scales to enforce controlled mixing rates while maintaining the target Hölder regularity and the divergence-free constraint.

If this is right

  • The dissipation rate stays positive and independent of time along the sequence of diffusivities.
  • The scalar solution gains higher time regularity than the classical Yaglom scaling predicts.
  • Anomalous dissipation holds uniformly over the entire low-regularity class of initial data.
  • The construction saturates the scaling relation β₀ + 2α₀ < 1 at any velocity regularity strictly below 1/3.

Where Pith is reading between the lines

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

  • The same layering technique could be tested on active-scalar equations or forced transport problems that share the linear advection structure.
  • Numerical experiments that embed the constructed velocity fields into a discrete advection-diffusion solver could directly measure the observed dissipation rate versus the predicted positive lower bound.
  • The complementary regularity condition β₀ + 2α₀ < 1 offers a concrete target for designing adaptive mesh refinements that preserve dissipation anomalies in computational mixing studies.
  • If the method extends to the inviscid limit, it would supply explicit examples of non-unique weak solutions for related transport equations at the same regularity level.

Load-bearing premise

The iterated homogenization procedure must produce velocity fields whose mixing properties simultaneously meet the Hölder regularity bound for the velocity, force positive energy dissipation for every low-regularity initial datum, and keep the scalar solution inside the complementary Hölder class.

What would settle it

A direct numerical or analytic computation of the integrated energy dissipation rate for one of the constructed solutions showing that this rate tends to zero as q tends to infinity for some admissible initial datum would disprove the central claim.

read the original abstract

For any $\beta_0<1/3$ we construct divergence free vector fields in $ C_{x,t}^{\beta_0}$ and a sequence of diffusivities $\kappa_q \searrow 0$ such that, for an arbitrary initial datum from a low regularity class, the classical solution $\rho_q$ to the advection-diffusion equation exhibits anomalous dissipation along the sequence $\kappa_q$. At the same time $\rho_q$ remains uniformly bounded in $C_t^{0} C_x^{\alpha_0}$, where $\beta_0 + 2\alpha_0<1$. Our result confirms a conjecture of Armstrong and Vicol \cite{ArmstrongVicol} and shows sharpness of the Obukhov-Corrsin threshold within the context of iterated homogenization. Our construction confirms time-homogeneity of the dissipation anomaly, as required in turbulence theory, and as a consequence we also obtain better time regularity for the scalar $\rho_q$ than the classical prediction of Yaglom.

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 / 3 minor

Summary. The manuscript constructs, for any β₀ < 1/3, divergence-free vector fields in the space C^{β₀}_{x,t} together with a sequence of diffusivities κ_q ↘ 0. For arbitrary initial data from a low-regularity class, the classical solutions ρ_q of the advection-diffusion equation are shown to exhibit anomalous dissipation along the sequence κ_q while remaining uniformly bounded in C^0_t C^{α₀}_x under the relation β₀ + 2α₀ < 1. The construction confirms the conjecture of Armstrong and Vicol, establishes sharpness of the Obukhov-Corrsin threshold in the iterated-homogenization setting, and yields time-homogeneous anomalous dissipation together with improved time regularity for ρ_q beyond the classical Yaglom prediction.

Significance. If the construction is valid, the result is significant because it supplies an explicit, divergence-free velocity field achieving anomalous scalar dissipation at the conjectured regularity threshold β₀ < 1/3. The iterated-homogenization method simultaneously controls the Hölder norms of the velocity and the scalar while preserving the divergence-free condition, thereby confirming a key open conjecture and aligning the mathematical construction with the time-homogeneity requirement of turbulence theory. The improved temporal regularity obtained for ρ_q is a further concrete contribution to the study of mixing in low-regularity advection-diffusion problems.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'arbitrary initial datum from a low regularity class' is left unspecified; stating the precise space (e.g., C^γ or B^γ_{p,q}) would clarify the scope of the existence result.
  2. [§1] §1 (Introduction): a short schematic diagram or numbered list summarizing the main steps of the iterated homogenization (scale selection, corrector construction, divergence-free projection) would help readers follow the technical argument.
  3. [§3] §3 (Construction): the uniform bound on the Hölder norm of ρ_q is stated to follow from the relation β₀ + 2α₀ < 1, but the precise interpolation or embedding inequality used to pass from the velocity regularity to the scalar bound should be recalled explicitly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The referee summary accurately reflects the main results: the construction of divergence-free C^{β₀} vector fields for any β₀ < 1/3, the sequence κ_q ↘ 0 producing anomalous dissipation for arbitrary low-regularity initial data, the uniform C^0_t C^{α₀}_x bound under β₀ + 2α₀ < 1, confirmation of the Armstrong-Vicol conjecture, sharpness of the Obukhov-Corrsin threshold in the iterated-homogenization setting, time-homogeneous dissipation, and the improved temporal regularity of ρ_q beyond the Yaglom prediction.

Circularity Check

0 steps flagged

No significant circularity; explicit construction confirming external conjecture

full rationale

The paper delivers an explicit constructive existence result via iterated homogenization, producing divergence-free velocities in the stated Hölder class together with a diffusivity sequence that forces anomalous dissipation for arbitrary low-regularity initial data while preserving the scalar bound under the relation β0 + 2α0 < 1. The central claim is presented as confirming an external conjecture of Armstrong and Vicol; no load-bearing equation reduces the dissipation rate, the Hölder exponents, or the time-homogeneity property to a fitted parameter, a self-defined quantity, or a prior result by the same authors. The derivation chain consists of the construction itself rather than any self-referential closure, renaming of known patterns, or uniqueness theorem imported from the authors' own work. The result is therefore self-contained against external benchmarks and exhibits no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard existence theory for advection-diffusion equations and on the technical feasibility of iterated homogenization to enforce both the velocity regularity and the dissipation anomaly; no free parameters or new entities are introduced beyond the constructed fields themselves.

axioms (2)
  • standard math For any divergence-free velocity field in C^{β0} and diffusivity κ>0 the advection-diffusion equation admits a unique classical solution for initial data in a low-regularity class.
    Invoked when the authors refer to the classical solution ρ_q.
  • domain assumption Iterated homogenization can be arranged to produce divergence-free fields with prescribed Hölder regularity and mixing rates that induce anomalous dissipation while preserving the scalar bound β0 + 2α0 <1.
    This is the core technical tool whose validity is required for the construction to succeed.

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