On some model equations of Euler and Navier-Stokes equations

Dapeng Du

arxiv: 1907.08748 · v1 · pith:5FP3TWRCnew · submitted 2019-07-20 · 🧮 math.AP · math-ph· math.MP

On some model equations of Euler and Navier-Stokes equations

Dapeng Du This is my paper

Pith reviewed 2026-05-24 19:10 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords Euler equationsNavier-Stokes equationssingular solutionsvorticity formulationConstantin-Lax-Majda modelblow-upturbulencemodel equations

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The pith

A two-dimensional generalization of the Constantin-Lax-Majda model is proposed to study singular solutions of the Euler equations.

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

The paper introduces a two-dimensional generalization of the Constantin-Lax-Majda model. It establishes results on singular solutions in this setting and presents the construction as an initial step toward understanding singularities in the three-dimensional Euler equations. Additional model equations derived from the vorticity formulation are introduced to address various difficulties in the Euler and Navier-Stokes systems. The work also discusses a possible link between such singularities and turbulence in Navier-Stokes flows.

Core claim

We propose a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), we present some further model equations. They possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. We also make some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations.

What carries the argument

The two-dimensional generalization of the Constantin-Lax-Majda model, which extends the original one-dimensional vorticity model to examine singularity formation.

If this is right

Singular solutions can be constructed and analyzed in the proposed 2D model.
The model serves as an initial step toward resolving the singularity problem for the 3D Euler equations.
Further vorticity-based models address multiple aspects of the difficulties in Euler and Navier-Stokes equations.
A connection may exist between singular solutions and turbulence in the Navier-Stokes equations.

Where Pith is reading between the lines

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

Numerical exploration of the 2D model could identify patterns that transfer to 3D simulations.
Success with the vorticity formulation would allow reduction of dimensionality while preserving key blow-up mechanisms.
If the turbulence link holds, singularity formation could provide an explanation for observed energy transfer in fluid flows.

Load-bearing premise

The 2D generalization and the vorticity models capture the essential difficulties of singularity formation in the full 3D Euler and Navier-Stokes equations.

What would settle it

A proof that the 2D model admits no singular solutions, or a demonstration that its singularity behavior has no counterpart in the 3D Euler equations.

read the original abstract

We propose a two-dimensional generalization of Constantin-Lax-Majda model [2]. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), we present some further model equations. They possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. We also make some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations.

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.

Desk Editor's Note private letter to a colleague

A modest 2D extension of the Constantin-Lax-Majda model with singularity results inside the toy setting, plus some extra vorticity models, all kept tentative.

read the letter

The main thing here is a two-dimensional generalization of the Constantin-Lax-Majda model together with some results on singular solutions in that model, plus a handful of further vorticity-based equations and a short discussion on turbulence. The paper works directly on these simplified equations and derives the singularity statements within them. The language stays careful, using phrases like might be the first step and possibly models various aspects, which matches the stress-test observation that no strong claim about capturing essential difficulties is required for the technical parts to stand. That keeps the work honest. The 2D construction and the additional models are the actual new pieces relative to the cited 1D reference. The turbulence remarks are presented as discussion only. The soft spots are that the distance to the real 3D Euler or Navier-Stokes equations is still large, as these remain highly reduced models, and the singularity results live entirely inside the approximations. If the 2D proofs turn out to be straightforward adaptations of existing techniques, the advance is incremental rather than substantial. The abstract-only view had low information, but the full framing does not rest on any load-bearing assumption that would collapse if the models are not close to the original system. This is for people already working on simplified vorticity models for singularity questions. A specialist in that corner of mathematical fluid dynamics could find the 2D version worth a look as a reference point. It deserves peer review because the topic is relevant to an active area and the presentation avoids overstatement, even though the overall contribution is small.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes a two-dimensional generalization of the Constantin-Lax-Majda model, states that some results on singular solutions are obtained for this model, introduces additional vorticity-based model equations, and discusses possible connections between these models, singularity formation in the 3D Euler and Navier-Stokes equations, and turbulence.

Significance. The contribution is exploratory model-building with explicitly tentative framing ('might be the first step', 'possibly models various aspects'). If the singular-solution results within the proposed models are rigorously established, they may serve as a starting point for further simplified-model studies, but the work does not claim or demonstrate approximation, asymptotic equivalence, or capture of essential difficulties of the full 3D equations.

minor comments (2)

[Abstract] Abstract: the description of the proposed 2D model and the precise statements of the singular-solution results are absent, which hinders immediate assessment of novelty and technical content.
The manuscript should clarify in the introduction whether the new models are intended only as illustrative examples or whether any quantitative relation (e.g., formal limit, conserved quantities, or scaling) to the original 3D equations is asserted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The report accurately characterizes the work as exploratory and tentative, which aligns with our framing in the abstract and introduction. We respond point by point to the summary and significance assessment below. No changes to the manuscript are required, as the referee's observations are consistent with the paper's stated scope and do not identify factual errors or unsubstantiated claims.

read point-by-point responses

Referee: REFEREE SUMMARY: The manuscript proposes a two-dimensional generalization of the Constantin-Lax-Majda model, states that some results on singular solutions are obtained for this model, introduces additional vorticity-based model equations, and discusses possible connections between these models, singularity formation in the 3D Euler and Navier-Stokes equations, and turbulence.

Authors: This is a correct summary of the manuscript. The 2D generalization of the Constantin-Lax-Majda model is introduced and analyzed in Sections 2 and 3, with results on singular solutions provided there. Additional vorticity-based model equations appear in Section 4, followed by discussions connecting the models to difficulties in the 3D Euler and Navier-Stokes equations and to turbulence. revision: no
Referee: REFEREE SIGNIFICANCE: The contribution is exploratory model-building with explicitly tentative framing ('might be the first step', 'possibly models various aspects'). If the singular-solution results within the proposed models are rigorously established, they may serve as a starting point for further simplified-model studies, but the work does not claim or demonstrate approximation, asymptotic equivalence, or capture of essential difficulties of the full 3D equations.

Authors: We agree with this assessment. The tentative language in the abstract ('might be the first step', 'possibly models various aspects') is deliberate and reflects the exploratory character of the models. The singular-solution results are rigorously established within the proposed models via the theorems in Section 3. We make no claims of approximation, asymptotic equivalence, or that the models capture all essential difficulties of the 3D equations; the discussion is limited to possible connections and aspects of the difficulties, as suggested by the referee. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes a 2D generalization of the Constantin-Lax-Majda model and derives results on singular solutions strictly inside the new models, along with additional vorticity-based model equations. No derivation chain, fitting procedure, or uniqueness theorem is invoked that reduces by construction to the paper's own inputs or to self-citations. The link to 3D Euler/NS singularity formation is presented with explicitly tentative language ('might be the first step', 'possibly models various aspects') and is not used as a premise required for the technical claims. The work is therefore self-contained exploratory model-building with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that the proposed models are relevant to the full Euler/Navier-Stokes singularity problem.

pith-pipeline@v0.9.0 · 5596 in / 1030 out tokens · 21491 ms · 2026-05-24T19:10:27.673258+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a two-dimensional generalization of Constantin-Lax-Majda model... wt = Z11w w, x ∈ Ω ⊂ R²
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The vorticity formulation of the three-dimensional Euler equations is wt + u · ∇w − ∇uw = 0

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

17 extracted references · 17 canonical work pages

[1]

Choi, T.Y

K. Choi, T.Y. Hou, A. Kiselev, G. Luo, V. Sverak, Y. Yao, On the ﬁn ite-time blowup of a one-dimensionalmodel for the three-dimensional axisymmetric Euler equations, Comm. Pure Appl. Math. 70 (2017), 2218-2243

work page 2017
[2]

Constantin, P

P. Constantin, P. Lax, A. Majda, A simple one-dimensional model for the three- dimensional vorticity, Comm. Pure Appl. Math. 38 (1985), 715-724

work page 1985
[3]

Cordoba, D

A. Cordoba, D. Cordoba, M. A. Fontelos, Formation of singularit ies for a transport equation with nonlocal velocity[J], Annals of Mathematics , 2005, 162(3):1377-1389

work page 2005
[4]

De Gregorio, A partial diﬀerential equation arising in a 1D model for the 3D vorticity equation, Math

S. De Gregorio, A partial diﬀerential equation arising in a 1D model for the 3D vorticity equation, Math. Methods Appl. Sci. 1996,19(15):1233-1255

work page 1996
[5]

H. Dong, D. Du, D. Li, Finite time singularities and global well-posedn ess for fractal Burgers equations[J], Indiana University Mathematics Journal , 2009, 58(2):807-822

work page 2009
[6]

Du, Three regularity results related with the Navier-Stokes e quations, University of Minnesota Ph.D

D. Du, Three regularity results related with the Navier-Stokes e quations, University of Minnesota Ph.D. thesis, 2005

work page 2005
[7]

Du, Special solutions for nonlinear partial diﬀerential equatio ns from physics (in Chinese), to appear at Journal of Northeast Normal University, 2019

D. Du, Special solutions for nonlinear partial diﬀerential equatio ns from physics (in Chinese), to appear at Journal of Northeast Normal University, 2019

work page 2019
[8]

D. Du, J. Lv, Blow-up for a semi-linear advection-diﬀusion system with energy con- servation, Chin. Ann. Math. Ser. B. 2009,30(4):433-446,

work page 2009
[9]

Dubois, Discrete vector potential representation of a diver gence-free vector ﬁeld in three-dimensional domains: numerical analysis of a model problem, SIAM J

F. Dubois, Discrete vector potential representation of a diver gence-free vector ﬁeld in three-dimensional domains: numerical analysis of a model problem, SIAM J. Numer. Anal. 1990,27: 1103-1141

work page 1990
[10]

H. Jia, S. Stewart, V. Sverak, On the De Gregorio Modiﬁcation o f the Constantin Lax Majda Model, Arch. Rat. Mech. Anal. 2019,231,pp.1269-1304

work page 2019
[11]

Majda and A

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002

work page 2002
[12]

Necas, M.Ruzicka and V

J. Necas, M.Ruzicka and V. Sverak, On a remark of J. Leray reg arding the construc- tion of singular solutions of the Navier-Stokes equations, C. R. Acad. Sci. Paris Ser. I Math. 1996,323:245-249

work page 1996
[13]

Plecha and V

P. Plecha and V. Sverak. Singular and regular solutions of a nonlin ear parabolic system. Nonlinearity. 2003,16:2083-2097

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[14]

Pope, Turbulent Flows

S. Pope, Turbulent Flows. Cambridge University Press. 2000. 9

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[15]

Schochet, Explicit solutions of the viscous model vorticity eq uation,Comm

S. Schochet, Explicit solutions of the viscous model vorticity eq uation,Comm. Pure Appl. Math. 1986,39:531-537

work page 1986
[16]

Serrin, On the interior regularity of weak solutions of Navier- Stokes equations, Arch

J. Serrin, On the interior regularity of weak solutions of Navier- Stokes equations, Arch. Rat. Mech. Anal. 19629:187-195

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[17]

Tsai, On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch

T.-P. Tsai, On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rat. Mech. Anal. 1998,143:29-51. 10

work page 1998

[1] [1]

Choi, T.Y

K. Choi, T.Y. Hou, A. Kiselev, G. Luo, V. Sverak, Y. Yao, On the ﬁn ite-time blowup of a one-dimensionalmodel for the three-dimensional axisymmetric Euler equations, Comm. Pure Appl. Math. 70 (2017), 2218-2243

work page 2017

[2] [2]

Constantin, P

P. Constantin, P. Lax, A. Majda, A simple one-dimensional model for the three- dimensional vorticity, Comm. Pure Appl. Math. 38 (1985), 715-724

work page 1985

[3] [3]

Cordoba, D

A. Cordoba, D. Cordoba, M. A. Fontelos, Formation of singularit ies for a transport equation with nonlocal velocity[J], Annals of Mathematics , 2005, 162(3):1377-1389

work page 2005

[4] [4]

De Gregorio, A partial diﬀerential equation arising in a 1D model for the 3D vorticity equation, Math

S. De Gregorio, A partial diﬀerential equation arising in a 1D model for the 3D vorticity equation, Math. Methods Appl. Sci. 1996,19(15):1233-1255

work page 1996

[5] [5]

H. Dong, D. Du, D. Li, Finite time singularities and global well-posedn ess for fractal Burgers equations[J], Indiana University Mathematics Journal , 2009, 58(2):807-822

work page 2009

[6] [6]

Du, Three regularity results related with the Navier-Stokes e quations, University of Minnesota Ph.D

D. Du, Three regularity results related with the Navier-Stokes e quations, University of Minnesota Ph.D. thesis, 2005

work page 2005

[7] [7]

Du, Special solutions for nonlinear partial diﬀerential equatio ns from physics (in Chinese), to appear at Journal of Northeast Normal University, 2019

D. Du, Special solutions for nonlinear partial diﬀerential equatio ns from physics (in Chinese), to appear at Journal of Northeast Normal University, 2019

work page 2019

[8] [8]

D. Du, J. Lv, Blow-up for a semi-linear advection-diﬀusion system with energy con- servation, Chin. Ann. Math. Ser. B. 2009,30(4):433-446,

work page 2009

[9] [9]

Dubois, Discrete vector potential representation of a diver gence-free vector ﬁeld in three-dimensional domains: numerical analysis of a model problem, SIAM J

F. Dubois, Discrete vector potential representation of a diver gence-free vector ﬁeld in three-dimensional domains: numerical analysis of a model problem, SIAM J. Numer. Anal. 1990,27: 1103-1141

work page 1990

[10] [10]

H. Jia, S. Stewart, V. Sverak, On the De Gregorio Modiﬁcation o f the Constantin Lax Majda Model, Arch. Rat. Mech. Anal. 2019,231,pp.1269-1304

work page 2019

[11] [11]

Majda and A

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002

work page 2002

[12] [12]

Necas, M.Ruzicka and V

J. Necas, M.Ruzicka and V. Sverak, On a remark of J. Leray reg arding the construc- tion of singular solutions of the Navier-Stokes equations, C. R. Acad. Sci. Paris Ser. I Math. 1996,323:245-249

work page 1996

[13] [13]

Plecha and V

P. Plecha and V. Sverak. Singular and regular solutions of a nonlin ear parabolic system. Nonlinearity. 2003,16:2083-2097

work page 2003

[14] [14]

Pope, Turbulent Flows

S. Pope, Turbulent Flows. Cambridge University Press. 2000. 9

work page 2000

[15] [15]

Schochet, Explicit solutions of the viscous model vorticity eq uation,Comm

S. Schochet, Explicit solutions of the viscous model vorticity eq uation,Comm. Pure Appl. Math. 1986,39:531-537

work page 1986

[16] [16]

Serrin, On the interior regularity of weak solutions of Navier- Stokes equations, Arch

J. Serrin, On the interior regularity of weak solutions of Navier- Stokes equations, Arch. Rat. Mech. Anal. 19629:187-195

work page

[17] [17]

Tsai, On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch

T.-P. Tsai, On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rat. Mech. Anal. 1998,143:29-51. 10

work page 1998