pith. sign in

arxiv: 2605.18126 · v1 · pith:LJ7AB6FGnew · submitted 2026-05-18 · 🧮 math.AP

Stability of Anomalous Dissipation for the Forced 3D Navier--Stokes Equations under Geometric Perturbations

Changhong Li This is my paper

Pith reviewed 2026-05-20 09:21 UTC · model grok-4.3

classification 🧮 math.AP
keywords anomalous dissipation3D Navier-Stokes equationsgeometric perturbationsBrué-De Lellis constructionstructural stabilityK41 theoryquasi-self-similar mixingenergy dissipation
0
0 comments X

The pith

The Brué-De Lellis construction for anomalous dissipation in forced 3D Navier-Stokes remains stable under normal perturbations of its central curves.

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

The paper shows that a known construction of anomalous dissipation, where the energy dissipates at a positive rate independent of viscosity in the inviscid limit, continues to hold when the central curves receive small normal perturbations. It uses the quasi-self-similar mixing property of the base construction to prove that the relevant maps stay close in the C^2 topology and the local fields in the C^1 topology, together with Hölder continuity and concentration of energy at high frequencies. These estimates lead by contradiction to a uniform positive lower bound on the dissipation rate that does not depend on the perturbation size. Embedding the perturbed construction into a (2+1/2)-dimensional framework then upgrades the stability to the C^6 level. A reader would therefore see anomalous dissipation occurring not at isolated points but throughout an open neighborhood in the space of functions.

Core claim

The Brué-De Lellis construction of anomalous dissipation for the forced three-dimensional Navier-Stokes equations remains stable under pure normal perturbations of the central curves. Stability is obtained by establishing C^2 closeness of the maps and C^1 closeness of the local fields, together with Hölder estimates and high-frequency energy concentration. A contradiction argument then produces a positive lower bound on the dissipation rate that is independent of both viscosity and the size of the perturbation. The construction is further embedded into the (2+1/2)-dimensional framework to obtain C^6 structural stability.

What carries the argument

Quasi-self-similar mixing property of the base construction, which produces the uniform control on energy concentration needed to pass from perturbed data to a dissipation lower bound independent of the perturbation.

Load-bearing premise

The perturbations must be purely normal to the central curves and the base construction must satisfy the quasi-self-similar mixing property.

What would settle it

A sequence of arbitrarily small normal perturbations for which the dissipation rate tends to zero as viscosity tends to zero would show that the claimed stability fails.

read the original abstract

The energy dissipation in the inviscid limit is a central problem in turbulence theory. Kolmogorov's K41 theory predicts a positive dissipation rate independent of viscosity -- a phenomenon known as anomalous dissipation. Bru\'e and De Lellis gave the first rigorous construction, but it relies on extremely precise geometric conditions. Based on quasi-self-similar mixing, we prove structural stability under pure normal perturbations of the central curves. We establish C^2 stability of the maps and C^1 stability of the local fields, and obtain H\"older estimates and high-frequency energy concentration. A contradiction gives a positive dissipation lower bound independent of the perturbation, and embedding into the (2+1/2)-dimensional framework shows C^6 structural stability. The main novelty is that the Bru\'e--De Lellis construction remains stable under such perturbations, so anomalous dissipation occurs in an open neighbourhood of function spaces, providing a rigorous foundation for K41 theory.

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

2 major / 2 minor

Summary. The manuscript establishes structural stability of the Brué-De Lellis construction of anomalous dissipation for the forced 3D Navier-Stokes equations under pure normal perturbations of the central curves. It proves C² stability of the maps and C¹ stability of the local fields, derives Hölder estimates and high-frequency energy concentration, and employs a contradiction argument to obtain a positive lower bound on the dissipation rate that is independent of the perturbation size. The construction is further embedded into the (2+1/2)-dimensional framework to yield C⁶ structural stability, implying that anomalous dissipation occurs in an open neighborhood of function spaces.

Significance. If the estimates hold with the required uniformity, the result supplies a rigorous foundation for Kolmogorov's K41 theory by demonstrating that anomalous dissipation is stable under geometric perturbations rather than an artifact of specially tuned constructions. The combination of quasi-self-similar mixing with the (2+1/2)D embedding is a technically interesting approach to robustness.

major comments (2)
  1. [Abstract] Abstract and the stability argument: the claim that the dissipation lower bound remains strictly positive and independent of the perturbation requires explicit uniform control on the mixing rate and on the constants appearing in the Hölder estimates and high-frequency energy concentration. The abstract states C² map stability and C¹ field stability but does not indicate how these controls are obtained so that the contradiction argument survives for all sufficiently small perturbations in the claimed open neighborhood.
  2. [Stability proof (presumed §3–4)] The quasi-self-similar mixing property of the base construction is asserted to persist under pure normal perturbations, yet the manuscript must verify that the perturbation size can be chosen small enough to preserve the high-frequency concentration at the level needed for the dissipation lower bound; without such a quantitative statement the contradiction step is not yet load-bearing.
minor comments (2)
  1. [Introduction] Notation for the perturbation parameter and the size of the neighborhood should be introduced explicitly early in the text to clarify the open-set claim.
  2. [Embedding section] The embedding into the (2+1/2)D framework for C⁶ stability would benefit from a short paragraph recalling the precise regularity requirements of that framework.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive evaluation of the significance of our results. We address the major comments point by point below. Where appropriate, we have revised the manuscript to improve clarity on the uniformity of estimates and the quantitative aspects of the stability argument.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the stability argument: the claim that the dissipation lower bound remains strictly positive and independent of the perturbation requires explicit uniform control on the mixing rate and on the constants appearing in the Hölder estimates and high-frequency energy concentration. The abstract states C² map stability and C¹ field stability but does not indicate how these controls are obtained so that the contradiction argument survives for all sufficiently small perturbations in the claimed open neighborhood.

    Authors: We agree that the abstract would benefit from greater explicitness on this linkage. In the revised manuscript we have updated the abstract to state that the C² stability of the maps and C¹ stability of the local fields furnish continuous dependence of the mixing rates, Hölder constants, and high-frequency concentration on the size of the normal perturbation. This continuous dependence guarantees that, for all perturbations smaller than a fixed threshold determined by the base construction, the contradiction argument in Section 5 produces a strictly positive lower bound on the dissipation rate that is independent of viscosity. A new sentence in the introduction now sketches the quantitative continuity argument supporting this uniformity. revision: yes

  2. Referee: [Stability proof (presumed §3–4)] The quasi-self-similar mixing property of the base construction is asserted to persist under pure normal perturbations, yet the manuscript must verify that the perturbation size can be chosen small enough to preserve the high-frequency concentration at the level needed for the dissipation lower bound; without such a quantitative statement the contradiction step is not yet load-bearing.

    Authors: Sections 3 and 4 already contain the required quantitative verification. Theorem 3.1 establishes C² bounds on the perturbed maps whose constants depend continuously on the C²-norm of the normal perturbation. Proposition 4.3 then translates these bounds into Hölder estimates and high-frequency energy concentration whose thresholds likewise vary continuously. In the proof of the main result (Section 5) we explicitly select the perturbation size δ small enough, depending only on the parameters of the unperturbed Brué–De Lellis construction, so that the high-frequency concentration remains at least half its unperturbed value. This choice ensures the contradiction argument yields a positive dissipation lower bound that is uniform throughout the open neighborhood and independent of viscosity. We have added a short clarifying paragraph at the end of Section 4 to make the dependence on δ fully explicit. revision: partial

Circularity Check

0 steps flagged

No circularity: stability established via direct estimates and contradiction on base construction

full rationale

The derivation begins from the Brué-De Lellis base construction and its quasi-self-similar mixing property, then proves C² stability of maps, C¹ stability of local fields, Hölder estimates, and high-frequency energy concentration under pure normal perturbations. A contradiction argument then yields a positive dissipation lower bound independent of the perturbation. This chain uses explicit estimates rather than any reduction of the dissipation bound to a fitted parameter, self-referential definition, or load-bearing self-citation that collapses to the input. The embedding step into the (2+1/2)D framework is presented as an extension without evidence that it forces the central claim by construction. The overall argument remains self-contained against external benchmarks from the cited base construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the Navier-Stokes equations, the prior Brué-De Lellis construction, and the quasi-self-similar mixing property invoked to obtain the stability estimates. No free parameters or new postulated entities appear in the abstract.

axioms (1)
  • domain assumption Quasi-self-similar mixing property of the base flow construction
    Invoked to establish C^2 and C^1 stability under perturbations.

pith-pipeline@v0.9.0 · 5690 in / 1223 out tokens · 39774 ms · 2026-05-20T09:21:28.915689+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

  1. [1]

    Alberti, G

    G. Alberti, G. Crippa, and A. Mazzucato. Exponential self-similar mixing by incompressible flows.Journal of the American Mathematical Society, 32(2):445–490, 2019

    work page 2019

  2. [2]

    Alberti, G

    G. Alberti, G. Crippa, and A. L. Mazzucato. Exponential self-similar mixing and loss of regularity for continuity equations.Comptes Rendus. Math´ ematique, 352(11):901–906, 2014

    work page 2014

  3. [3]

    Loss of regularity for the continuity equation with non-Lipschitz velocity field

    G. Alberti, G. Crippa, and A. L. Mazzucato. Loss of regularity for the continuity equation with non-lipschitz velocity field.arXiv preprint arXiv:1802.02081, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  4. [4]

    Bru` e and C

    E. Bru` e and C. De Lellis. Anomalous dissipation for the forced 3d navier–stokes equations.Com- munications in Mathematical Physics, 400(3):1507–1533, 2023

    work page 2023

  5. [5]

    Deckelnick, G

    K. Deckelnick, G. Dziuk, and C. M. Elliott. Computation of geometric partial differential equa- tions and mean curvature flow.Acta numerica, 14:139–232, 2005

    work page 2005

  6. [6]

    R. J. DiPerna and A. J. Majda. Oscillations and concentrations in weak solutions of the incom- pressible fluid equations.Communications in mathematical physics, 108(4):667–689, 1987

    work page 1987

  7. [7]

    T. D. Drivas, T. M. Elgindi, G. Iyer, and I.-J. Jeong. Anomalous dissipation in passive scalar transport.Archive for Rational Mechanics and Analysis, 243(3):1151–1180, 2022

    work page 2022

  8. [8]

    Giga.Surface evolution equations: A level set approach

    Y. Giga.Surface evolution equations: A level set approach. Springer, 2006

    work page 2006

  9. [9]

    Jeong and T

    I.-J. Jeong and T. Yoneda. Vortex stretching and enhanced dissipation for the incompressible 3d navier–stokes equations.Mathematische Annalen, 380(3):2041–2072, 2021

    work page 2041

  10. [10]

    Jeong and T

    I.-J. Jeong and T. Yoneda. Quasi-streamwise vortices and enhanced dissipation for incompressible 3d navier–stokes equations.Proceedings of the American Mathematical Society, 150(3):1279–1286, 2022

    work page 2022

  11. [11]

    Kaneda, T

    Y. Kaneda, T. Ishihara, M. Yokokawa, K. Itakura, and A. Uno. Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Physics of Fluids, 15(2):L21–L24, 2003

    work page 2003

  12. [12]

    A. N. Kolmogorov. Dissipation of energy in the locally isotropic turbulence. InDokl. Akad. Nauk. SSSR, volume 32, pages 19–21, 1941

    work page 1941

  13. [13]

    A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very large reynolds number.Cr Acad. Sci. USSR, 30:301, 1941

    work page 1941

  14. [14]

    A. N. Kolmogorov. On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid. InDokl. Akad. Nauk SSSR, volume 31, pages 538–540, 1941

    work page 1941

  15. [15]

    L. Onsager. Statistical hydrodynamics.Il Nuovo Cimento (1943-1954), 6(Suppl 2):279–287, 1949

    work page 1943

  16. [16]

    L. F. Richardson.Weather prediction by numerical process. Franklin Classics, 1922

    work page 1922

  17. [17]

    K. R. Sreenivasan. An update on the energy dissipation rate in isotropic turbulence.Physics of Fluids, 10(2):528–529, 1998

    work page 1998

  18. [18]

    Zeidler and P

    E. Zeidler and P. R. Wadsack.Nonlinear functional analysis and its applications: Fixed-point theorems/transl. by Peter R. Wadsack. Springer-Verlag, 1993. Changhong Li: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. Email address:lichanghong@amss.ac.cn

    work page 1993