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arxiv: 2606.04218 · v1 · pith:BOTTHB2Mnew · submitted 2026-06-02 · 🧮 math.AP

Absence of local anomalous dissipation and local energy balance in 2D incompressible flows away from the boundary

Tobias Barker , Milton C. Lopes Filho , Helena J. Nussenzveig Lopes This is my paper

Pith reviewed 2026-06-28 08:51 UTC · model grok-4.3

classification 🧮 math.AP
keywords Navier-Stokes equationsanomalous dissipationinviscid limit2D incompressible flowlocal energy balanceOnsager critical spaces
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The pith

Uniform bounds in Onsager supercritical space make local anomalous dissipation vanish away from the boundary in 2D Navier-Stokes.

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

The paper shows that for 2D Navier-Stokes equations with no-slip boundaries, if the velocity is uniformly bounded in L^{1+}_t L^∞_{x,loc} with controlled initial data, then anomalous dissipation vanishes in the interior. This leads to convergence to an Euler solution where the large scale approximation satisfies a local energy balance. The proof uses localization via modulation and vorticity energy estimates without needing bounds on the pressure. Such a result matters because it identifies conditions under which the viscous term disappears locally in the inviscid limit for flows in bounded domains.

Core claim

We show that anomalous dissipation vanishes locally away from the boundary for solutions of the 2D Navier-Stokes equations with no-slip boundary condition if the velocity u^ν is uniformly bounded in the Onsager supercritical space L^{1+}_t L^∞_{x,loc} with appropriate initial conditions. This setting produces convergence to an Euler solution whose large scale approximation satisfies a local energy balance equation, without assuming uniform-in-viscosity bounds on the pressure.

What carries the argument

Localization via modulation together with vorticity energy estimates and L^2-based structure functions.

If this is right

  • The inviscid limit converges to an Euler solution in the interior.
  • The large scale approximation of the limit satisfies local energy balance.
  • These conclusions hold without uniform bounds on the pressure.
  • The result is specific to two-dimensional incompressible flows away from boundaries.

Where Pith is reading between the lines

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

  • Boundary layers are likely responsible for any persistent anomalous dissipation in 2D flows.
  • Similar localization techniques might apply to other dimensions or boundary conditions if adapted.
  • Testing the bound numerically in periodic domains or with slip boundaries could isolate interior behavior.

Load-bearing premise

The cited localization and vorticity estimate techniques apply directly to regions away from the no-slip boundary without change.

What would settle it

Finding a sequence of solutions with the stated L^{1+}_t L^∞ bound but positive local anomalous dissipation in the interior would disprove the vanishing result.

read the original abstract

For the 2D Navier-Stokes equations with no-slip boundary condition, we consider the issue of whether anomalous dissipation away from the boundary vanishes. In particular, we show that such vanishing occurs if $u^{\nu}$ is uniformily bounded in the Onsager supercritical space $L^{1+}_{t}L^{\infty}_{x,loc}$ with appropriate bounds on the initial conditions. Our method involves arguments from \cite{AD23} and \cite{CW23} involving localization via modulation, together with vorticity energy type estimates inspired by \cite{CFLS16} and estimates involving $L^2$-based structure functions inspired by \cite{DP25dissconc}. Next we show that the aforementioned setting produces convergence to an Euler solution with its large scale approximation satisfying a local energy balance equation. Notably, we do not assume any uniform-in-viscosity bounds on the pressure. The large scale approximation has been introduced in \cite{PGLR18} in the context of partial regularity of the 3D Navier-Stokes equations, yet to the best of our knowledge this is the first time it has been considered in the context of inviscid limits.

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 proves absence of local anomalous dissipation for 2D Navier-Stokes with no-slip boundary conditions away from the boundary, under the assumption that u^ν is uniformly bounded in the Onsager-supercritical space L^{1+}_t L^∞_{x,loc} (with suitable initial data). The argument adapts localization-via-modulation techniques from AD23 and CW23 together with vorticity-energy and L^2 structure-function estimates from CFLS16 and DP25dissconc. It further establishes convergence to an Euler solution whose large-scale approximation (in the sense of PGLR18) satisfies a local energy balance, without any uniform-in-viscosity pressure bounds.

Significance. If the localization closes, the result would extend inviscid-limit and anomalous-dissipation theory to bounded domains with physical boundary conditions while avoiding pressure control; the application of the large-scale approximation to the 2D inviscid limit is new. The explicit use of only local supercritical bounds is a potentially useful strengthening over global assumptions in prior works.

major comments (2)
  1. [Proof of the vanishing of local anomalous dissipation (adaptation of AD23/CW23 and CFLS16/DP25dissconc estimates)] The central claim that local anomalous dissipation vanishes rests on transferring the localization-via-modulation and vorticity estimates to an interior subdomain Ω' ⋐ Ω. The no-slip condition on ∂Ω generates nonlocal pressure contributions and possible vorticity boundary-layer effects that reach into Ω' at positive distance; the manuscript supplies only the local L^{1+}_t L^∞_{x,loc} bound and states that no uniform pressure control is assumed. It is therefore necessary to verify, in the derivation of the localized energy balance (the step that closes the dissipation estimate), that all resulting commutator and pressure terms are absorbed using solely the given local bound. If these terms are not explicitly cancelled or estimated, the vanishing result does not follow.
  2. [Convergence to Euler and local energy balance for the large-scale approximation] The subsequent convergence statement to an Euler solution with local energy balance for the large-scale approximation likewise depends on the same localized dissipation control. Any unaccounted boundary-induced error in the interior estimates would propagate into the limit and undermine the local energy balance claim.
minor comments (2)
  1. [Abstract] The abstract contains the typographical error "uniformily" (should be "uniformly").
  2. [Statement of main results] Notation for the interior subdomain Ω' and the precise meaning of the local bound (including the precise dependence on dist(Ω',∂Ω)) should be stated explicitly at the beginning of the main theorem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the opportunity to address these points. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Proof of the vanishing of local anomalous dissipation (adaptation of AD23/CW23 and CFLS16/DP25dissconc estimates)] The central claim that local anomalous dissipation vanishes rests on transferring the localization-via-modulation and vorticity estimates to an interior subdomain Ω' ⋐ Ω. The no-slip condition on ∂Ω generates nonlocal pressure contributions and possible vorticity boundary-layer effects that reach into Ω' at positive distance; the manuscript supplies only the local L^{1+}_t L^∞_{x,loc} bound and states that no uniform pressure control is assumed. It is therefore necessary to verify, in the derivation of the localized energy balance (the step that closes the dissipation estimate), that all resulting commutator and pressure terms are absorbed using solely the given local bound. If these terms are not explicitly cancelled or estimated, the vanishing result does not follow.

    Authors: In the derivation of the localized energy balance in Section 3, the cutoff function is chosen with support in Ω' at positive distance from ∂Ω. The modulation technique is applied within this interior region, and the commutator terms (including those arising from the pressure) are controlled via the local L^{1+}_t L^∞ bound combined with the L^2 structure-function estimates adapted from DP25dissconc. These estimates absorb the contributions without global pressure bounds because the distance to the boundary ensures that nonlocal effects remain controllable locally. The vorticity-energy estimates from CFLS16 are likewise localized to Ω', so boundary-layer influences do not enter the interior balance at the scales considered. We will add a short clarifying paragraph in the revised manuscript explicitly listing the absorption steps for these terms. revision: partial

  2. Referee: [Convergence to Euler and local energy balance for the large-scale approximation] The subsequent convergence statement to an Euler solution with local energy balance for the large-scale approximation likewise depends on the same localized dissipation control. Any unaccounted boundary-induced error in the interior estimates would propagate into the limit and undermine the local energy balance claim.

    Authors: The convergence argument and the local energy balance for the large-scale approximation (Section 4) are direct consequences of the vanishing of local anomalous dissipation proved in Section 3. Because the interior estimates close using only the given local bound (as outlined above), no boundary-induced errors remain to propagate into the inviscid limit. The large-scale approximation is applied on subdomains strictly inside Ω, where the uniform-in-viscosity control holds. revision: no

Circularity Check

0 steps flagged

No significant circularity; central claims adapt external estimates without reduction to inputs by construction

full rationale

The manuscript establishes vanishing of local anomalous dissipation under a local L^{1+}_t L^∞ bound by adapting localization-via-modulation from the cited works AD23 and CW23 together with vorticity and structure-function estimates from CFLS16 and DP25dissconc; these are treated as independent external inputs whose arguments are transferred to an interior subdomain. The subsequent convergence to an Euler solution whose large-scale approximation satisfies local energy balance is derived from that vanishing, again without redefining the target quantity in terms of itself or fitting parameters to the conclusion. No equation or step in the provided abstract or description reduces the claimed result to a self-citation chain or a fitted input renamed as prediction. The derivation therefore remains self-contained against the external benchmarks supplied by the citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard existence theory for 2D Navier-Stokes and on the direct applicability of localization and vorticity estimates from the four cited papers; no free parameters or new postulated entities appear.

axioms (2)
  • domain assumption Existence of solutions u^ν to the 2D Navier-Stokes system with no-slip boundary conditions in the relevant spaces
    Required to even state the family of solutions under consideration
  • ad hoc to paper Localization via modulation and the vorticity energy estimates extend to the interior away from the boundary without additional error terms
    The method section explicitly invokes arguments from AD23, CW23, CFLS16 and DP25dissconc

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Reference graph

Works this paper leans on

48 extracted references · 4 canonical work pages · 2 internal anchors

  1. [1]

    K. Abe. On the large timeL ∞-estimates of the Stokes semigroup in two-dimensional exterior domains.J. Differ. Equations, 300:337–355, 2021. 7

    2021

  2. [2]

    Abe and Y

    K. Abe and Y. Giga. Analyticity of the Stokes semigroup in spaces of bounded functions. Acta Math., 211(1):1–46, 2013. 7

    2013

  3. [3]

    Albritton and H

    D. Albritton and H. Dong. Regularity properties of passive scalars with rough divergence-free drifts.Arch. Ration. Mech. Anal., 247(5):44, 2023. Id/No 75. 1, 5, 10, 21

    2023

  4. [4]

    Bardos, E

    C. Bardos, E. S. Titi, and E. Wiedemann. Onsager’s conjecture with physical boundaries and an application to the vanishing viscosity limit.Commun. Math. Phys., 370(1):291–310, 2019. 5, 6

    2019

  5. [5]

    Bru` e, M

    E. Bru` e, M. Colombo, and A. Kumar. Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity. Preprint, arXiv:2408.07934 [math.AP] (2024), 2024. 3

    work page arXiv 2024

  6. [6]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, P. Isett, and L. Sz´ ekelyhidi Jr. Anomalous dissipation for 1/5- H¨ older Euler flows.Ann. Math. (2), 182(1):127–172, 2015. 2 30 T. BARKER ET AL

    2015

  7. [7]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, and L. Sz´ ekelyhidi Jr. Dissipative Euler flows with Onsager- critical spatial regularity.Commun. Pure Appl. Math., 69(9):1613–1670, 2016. 2

    2016

  8. [8]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, L. Sz´ ekelyhidi Jr., and V. Vicol. Onsager’s conjecture for ad- missible weak solutions.Commun. Pure Appl. Math., 72(2):229–274, 2019. 2

    2019

  9. [9]

    Buckmaster, N

    T. Buckmaster, N. Masmoudi, M. Novack, and V. Vicol.Intermittent convex integration for the 3D Euler equations, volume 217 ofAnn. Math. Stud.Princeton, NJ: Princeton University Press, 2023. 2

    2023

  10. [10]

    Buckmaster and V

    T. Buckmaster and V. Vicol. Convex integration and phenomenologies in turbulence.EMS Surv. Math. Sci., 6(1-2):173–263, 2019. 2

    2019

  11. [11]

    Buckmaster and V

    T. Buckmaster and V. Vicol. Convex integration constructions in hydrodynamics.Bull. Am. Math. Soc., New Ser., 58(1):1–44, 2021. 2

    2021

  12. [12]

    Chae and J

    D. Chae and J. Wolf. Localized blow-up criterion forC 1,α solutions to the 3D incompressible Euler equations.J. Math. Fluid Mech., 25(3):29, 2023. Id/No 68. 1, 5, 10

    2023

  13. [13]

    Chamorro, P.-G

    D. Chamorro, P.-G. Lemari´ e-Rieusset, and K. Mayoufi. The role of the pressure in the partial regularity theory for weak solutions of the Navier-Stokes equations.Arch. Ration. Mech. Anal., 228(1):237–277, 2018. 1, 6

    2018

  14. [14]

    Chen and J

    G.-Q. Chen and J. Glimm. Kolmogorov’s theory of turbulence and inviscid limit of the Navier- Stokes equations inR 3.Commun. Math. Phys., 310(1):267–283, 2012. 2

    2012

  15. [15]

    Cheskidov, P

    A. Cheskidov, P. Constantin, S. Friedlander, and R. Shvydkoy. Energy conservation and Onsager’s conjecture for the Euler equations.Nonlinearity, 21(6):1233–1252, 2008. 2

    2008

  16. [16]

    Cheskidov, M

    A. Cheskidov, M. C. Lopes Filho, H. J. Nussenzveig Lopes, and R. Shvydkoy. Energy conser- vation in two-dimensional incompressible ideal fluids.Commun. Math. Phys., 348(1):129–143,

  17. [17]

    1, 2, 3, 4, 5, 6, 16

  18. [18]

    Constantin, W

    P. Constantin, W. E, and E. S. Titi. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation.Commun. Math. Phys., 165(1):207–209, 1994. 2

    1994

  19. [19]

    Constantin, M

    P. Constantin, M. C. Lopes Filho, H. J. Nussenzveig Lopes, and V. Vicol. Vorticity measures and the inviscid limit.Arch. Ration. Mech. Anal., 234(2):575–593, 2019. 5

    2019

  20. [20]

    Constantin and V

    P. Constantin and V. Vicol. Remarks on high Reynolds numbers hydrodynamics and the inviscid limit.J. Nonlinear Sci., 28(2):711–724, 2018. 5

    2018

  21. [21]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi Jr. The Euler equations as a differential inclusion.Ann. Math. (2), 170(3):1417–1436, 2009. 2

    2009

  22. [22]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi Jr. Dissipative continuous Euler flows.Invent. Math., 193(2):377–407, 2013. 2

    2013

  23. [23]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi Jr. Dissipative Euler flows and Onsager’s conjecture.J. Eur. Math. Soc. (JEMS), 16(7):1467–1505, 2014. 2

    2014

  24. [24]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi Jr. Weak stability and closure in turbulence.Philosophi- cal Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 380(2218):20210091, 01 2022. 2

    2022

  25. [25]

    De Rosa and M

    L. De Rosa and M. Inversi. Dissipation in Onsager’s critical classes and energy conservation inBV∩L ∞ with and without boundary.Commun. Math. Phys., 405(1):34, 2024. Id/No 6. 2

    2024

  26. [26]

    De Rosa, M

    L. De Rosa, M. Inversi, and M. Nesi. Dissipation for codimension 1 singular structures in the incompressible Euler equations.Nonlinearity, 39(2):16, 2026. Id/No 025002. 2

    2026

  27. [27]

    Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation

    L. De Rosa, M. Latocca, and J. Park. Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation. Preprint, arXiv:2509.01268 [math.AP] (2025), 2025. 5

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  28. [28]

    De Rosa and J

    L. De Rosa and J. Park. No anomalous dissipation in two-dimensional incompressible fluids. SIAM J. Math. Anal., 57(5):5771–5790, 2025. 5

    2025

  29. [29]

    Dissipation concentration in two-dimensional fluids

    L. De Rosa and J. Park. Dissipation concentration in two-dimensional fluids. Preprint, arXiv:2508.01440 [math.AP] (2026), 2026. 1, 4, 6, 12, 13, 15

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  30. [30]

    T. D. Drivas and H. Q. Nguyen. Onsager’s conjecture and anomalous dissipation on domains with boundary.SIAM J. Math. Anal., 50(5):4785–4811, 2018. 5, 6

    2018

  31. [31]

    Duchon and R

    J. Duchon and R. Robert. Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations.Nonlinearity, 13(1):249–255, 2000. 6

    2000

  32. [32]

    T. M. Elgindi, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. Absence of anomalous dissipation for vortex sheets.J. Funct. Anal., 290(6):25, 2026. Id/No 111304. 5

    2026

  33. [33]

    G. L. Eyink. Energy dissipation without viscosity in ideal hydrodynamics. I: Fourier analysis and local energy transfer.Physica D, 78(3-4):222–240, 1994. 2 ABSENCE OF LOCAL ANOMALOUS DISSIPATION 31

    1994

  34. [34]

    V. Giri, H. Kwon, and M. Novack. TheL 3-based strong Onsager theorem. Preprint, arXiv:2305.18509 [math.AP] (2025). To appear in Annals of Mathematics., 2025. 2

    work page arXiv 2025

  35. [35]

    Giri and R.-O

    V. Giri and R.-O. Radu. The Onsager conjecture in 2D: a Newton-Nash iteration.Invent. Math., 238(2):691–768, 2024. 2

    2024

  36. [36]

    P. Isett. A proof of Onsager’s conjecture.Ann. Math. (2), 188(3):871–963, 2018. 2

    2018

  37. [37]

    F. Jin, S. Lanthaler, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. Sharp conditions for energy balance in two-dimensional incompressible ideal flow with external force.Nonlinearity, 38(7):46, 2025. Id/No 075007. 5

    2025

  38. [38]

    J. P. Kelliher. Observations on the vanishing viscosity limit.Trans. Am. Math. Soc., 369(3):2003–2027, 2017. 5

    2003

  39. [39]

    Koch and V

    H. Koch and V. A. Solonnikov.L q-estimates of the first-order derivatives of solutions to the nonstationary Stokes problem. InNonlinear problems in mathematical physics and related topics I. In honor of Professor O. A. Ladyzhenskaya, pages 203–218. New York, NY: Kluwer Academic/Plenum Publishers, 2002. 6

    2002

  40. [40]

    A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Translated from the Russian C. R. (Doklady) Acad. Sci. URSS (N.S.) 30, 301–305 (1941) by V. Levin.Proc. R. Soc. Lond., Ser. A, 434(1890):9–13, 1991. 2

    1941

  41. [41]

    A. N. Kolmogorov, V. Levin, J. C. R. Hunt, O. M. Phillips, and D. Williams. Dissipation of energy in the locally isotropic turbulence. Translated from the Russian C. R. (Doklady) Acad. Sci. URSS (N.S.) 32, 16–18 (1941) by V. Levin.Proc. R. Soc. Lond., Ser. A, 434(1890):15–17, 07 1991. 2

    1941

  42. [42]

    H. Kwon. The role of the pressure in the regularity theory for the Navier-Stokes equations. J. Differ. Equations, 357:1–31, 2023. 6, 8, 21, 22, 23

    2023

  43. [43]

    Lanthaler, S

    S. Lanthaler, S. Mishra, and C. Par´ es-Pulido. On the conservation of energy in two- dimensional incompressible flows.Nonlinearity, 34(2):1084–1135, 2021. 4, 5

    2021

  44. [44]

    M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, and M. Taylor. Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows.Bull. Braz. Math. Soc. (N.S.), 39(4):471–513, 2008. 6

    2008

  45. [45]

    M. C. Lopes Filho and H. J. Nussenzveig Lopes. Energy balance for forced two-dimensional incompressible ideal fluid flow†.Philosophical Transactions of the Royal Society A: Mathe- matical, Physical and Engineering Sciences, 380(2219):20210095, 01 2022. 5

    2022

  46. [46]

    D. S. McCormick, J. C. Robinson, and J. L. Rodrigo. Generalised Gagliardo-Nirenberg in- equalities using weak Lebesgue spaces and BMO.Milan J. Math., 81(2):265–289, 2013. 16

    2013

  47. [47]

    Novack and V

    M. Novack and V. Vicol. An intermittent Onsager theorem.Invent. Math., 233(1):223–323,

  48. [48]

    C. Seis, E. Wiedemann, and J. Wo´ znicki. Strong convergence of vorticities in the 2D viscosity limit on a bounded domain.J. Nonlinear Sci., 36(1):23, 2026. Id/No 22. 5, 7, 24

    2026