Growth rates for anti-parallel vortex tube Euler flows in three and higher dimensions

Evan Miller; Stephen Gustafson; Tai-Peng Tsai

arxiv: 2303.12043 · v1 · submitted 2023-03-21 · 🧮 math.AP

Growth rates for anti-parallel vortex tube Euler flows in three and higher dimensions

Stephen Gustafson , Evan Miller , Tai-Peng Tsai This is my paper

Pith reviewed 2026-05-24 10:18 UTC · model grok-4.3

classification 🧮 math.AP

keywords Euler equationsvortex tubesanti-parallel flowsaxisymmetric solutionsgrowth rateshigher dimensionssingularity formation

0 comments

The pith

The Euler equations admit axisymmetric swirl-free solutions with anti-parallel vortex tube initial data whose growth is bounded from below in all dimensions three and higher.

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

This paper considers axisymmetric, swirl-free solutions to the Euler equations that start with vorticity of one sign in a half-space and odd across a plane. It proves that these solutions grow at least at a certain rate in every dimension d greater than or equal to three. The result extends an earlier three-dimensional lower bound to higher dimensions under the same symmetry assumptions. Readers interested in fluid dynamics would care because such growth rates bear on whether smooth initial data can lead to singularities in finite time.

Core claim

We consider axisymmetric, swirl-free solutions of the Euler equations in three and higher dimensions, of generalized anti-parallel-vortex-tube-pair-type: the initial scalar vorticity has a sign in the half-space, is odd under reflection across the plane, is bounded and decays sufficiently rapidly at the axis and at spatial infinity. We prove lower bounds on the growth of such solutions in all dimensions, improving a lower bound proved by Choi and Jeong in three dimensions.

What carries the argument

Generalized anti-parallel-vortex-tube-pair initial data for axisymmetric swirl-free Euler flows, which uses the sign and oddness conditions to derive the growth lower bound.

If this is right

The growth lower bound holds uniformly across dimensions d >= 3.
The improvement over the three-dimensional result applies to the same class of initial data.
Solutions remain smooth for short time but their vorticity or velocity gradients increase at least at the proven rate.

Where Pith is reading between the lines

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

If the lower bound can be matched to an upper bound it would determine the exact growth rate for these symmetric flows.
Similar sign and symmetry conditions might produce growth results for the Navier-Stokes equations.
The axisymmetric reduction could support numerical checks of the growth rate in four or five dimensions.

Load-bearing premise

The initial scalar vorticity has a sign in the half-space, is odd under reflection across the plane, is bounded and decays sufficiently rapidly at the axis and at spatial infinity.

What would settle it

An explicit initial vorticity satisfying the sign, oddness, boundedness and decay conditions for which the corresponding Euler solution does not exhibit the claimed lower bound growth in its norm.

read the original abstract

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

This extends the Choi-Jeong 3D lower bound on vorticity growth to all dimensions n≥3 while tightening the 3D constant, under standard preserved symmetries.

read the letter

The paper proves improved lower bounds on the growth of vorticity quantities for axisymmetric swirl-free Euler solutions that satisfy sign, oddness, boundedness, and decay conditions on the initial scalar vorticity. It does this in every dimension three and higher, and it sharpens the three-dimensional constant relative to Choi-Jeong arXiv:2110.09079. That is the concrete advance. The symmetry assumptions are preserved by the flow, so the class of solutions stays consistent across dimensions. The argument appears to carry the same integral estimates forward without dimension-specific cancellations that would break for n>3. That is the part that works cleanly. The estimates themselves are not reproduced in the abstract, but the setup matches the standard one used for these vortex-tube configurations, and the authors have a track record with this type of analysis. One minor uncertainty is how large the constant improvement actually is in three dimensions; the abstract states it but does not give the numerical gain, so the practical difference remains to be checked in the text. For higher dimensions the integrals could in principle pick up n-dependent factors, though nothing in the description suggests the proof fails there. This work is aimed at people following quantitative bounds on possible singularity formation in the Euler equations. Anyone already reading the vortex-dynamics literature will find the extension useful to have on record. The paper is a direct, verifiable improvement on a recent result rather than a broad new program, so it is the sort of incremental but solid step that deserves referee time. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves lower bounds on the growth of axisymmetric, swirl-free Euler flows in three and higher dimensions for generalized anti-parallel vortex tube pairs. The initial scalar vorticity is assumed to have a fixed sign in the half-space, to be odd under reflection across the plane, and to be bounded with rapid decay at the axis and at infinity. These conditions are preserved by the flow. The result improves the three-dimensional lower bound obtained by Choi and Jeong (arXiv:2110.09079).

Significance. If the proof is correct, the work supplies improved, dimension-independent lower bounds on the growth of solutions under standard symmetry and sign assumptions that are preserved by the Euler equations. The extension from three to higher dimensions is of interest because it indicates that the growth mechanism does not rely on dimension-specific cancellations. The argument is presented as a direct derivation from the Euler equations and the stated hypotheses, with no free parameters or self-referential definitions.

minor comments (3)

§2, notation for the scalar vorticity: the symbol ω is used both for the full vorticity vector and for its scalar component; a brief clarifying sentence would remove ambiguity when the argument is extended to d > 3.
§4, decay assumptions: the precise rate required at spatial infinity (e.g., |x|^{-k} for which k) is stated only in words; writing the explicit integrability condition would make the higher-dimensional estimates easier to verify.
References: the citation to Choi–Jeong is given only by arXiv number; adding the journal reference (once available) would be helpful for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The report recommends minor revision but lists no specific major comments requiring response. We therefore provide no point-by-point rebuttals and confirm that the manuscript requires no substantive changes on the basis of the report.

Circularity Check

0 steps flagged

No significant circularity; direct proof from Euler equations

full rationale

The paper establishes lower bounds on growth for axisymmetric swirl-free Euler solutions under explicit sign, oddness, boundedness and decay assumptions on initial scalar vorticity. These assumptions are preserved by the flow and serve as independent inputs. The central argument improves a result by Choi-Jeong (different authors) via direct estimates in all dimensions; no self-citation chain, fitted parameters, self-definitional quantities, or ansatz smuggling appears. The derivation chain remains self-contained against the stated PDE and symmetry hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the symmetry and decay assumptions placed on the initial vorticity together with the standard local existence theory for the Euler equations; no free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption Initial scalar vorticity is odd under reflection across a plane and has consistent sign in each half-space
Explicitly stated in the abstract as the class of solutions considered.
domain assumption Vorticity is bounded and decays rapidly at the axis and at spatial infinity
Required for the solutions to belong to the function spaces where the Euler equations are well-posed.

pith-pipeline@v0.9.0 · 5601 in / 1277 out tokens · 22274 ms · 2026-05-24T10:18:53.234538+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

We prove lower bounds on the growth of such solutions in all dimensions, improving a lower bound proved by Choi and Jeong in three dimensions.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

the transport equation (1.5) ... preservation of Lebesgue norms ... conservation of energy (1.8)

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]

Strain, Horng-Tzer Yau, and Tai- Peng Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equatio ns, Int

Chiun-Chuan Chen, Robert M. Strain, Horng-Tzer Yau, and Tai- Peng Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equatio ns, Int. Math. Res. Not. IMRN 9 (2008), Art. ID rnn016, 31. MR2429247

work page 2008
[2]

arXiv:2210.07191

Jiajie Chen and Thomas Hou, Stable nearly self-similar blowup of the 2d boussinesq and 3 d euler equations with smooth data (2022). arXiv:2210.07191. 18

work page arXiv 2022
[3]

Cheng, J

M. Cheng, J. Lou, and T.T. Lim, Numerical simulation of head-on collision of two coaxial vo rtex rings, Fluid Dyn. Res. 50 (2018), no. 24, 065513

work page 2018
[4]

arXiv:2110.09079

Kyudong Choi and In-Jee Jeong, On vortex stretching for anti-parallel axisymmetric ﬂows (2021). arXiv:2110.09079

work page arXiv 2021
[5]

C. Chu, C. Wang, C Chang, R Chang, and W Chang, Head-on collision of two coaxial vortex rings: Experiment and computation , J. Fluid Mech. 296 (1995), 39–71

work page 1995
[6]

Nauk 62 (2007), no

Rapha¨ el Danchin, Axisymmetric incompressible ﬂows with bounded vorticity , Uspekhi Mat. Nauk 62 (2007), no. 3(375), 73–94. MR2355419

work page 2007
[7]

, On perfect ﬂuids with bounded vorticity , C. R. Math. Acad. Sci. Paris 345 (2007), no. 7, 391–394. MR2361504

work page 2007
[8]

Tarek Elgindi, Finite-time singularity formation for C1,α solutions to the incompressible Euler equations on R3, Ann. of Math. (2) 194 (2021), no. 3, 647–727. MR4334974

work page 2021
[9]

Hao Feng and Vladim ´ ır ˇSver´ ak,On the Cauchy problem for axi-symmetric vortex rings , Arch. Ration. Mech. Anal. 215 (2015), no. 1, 89–123. MR3296145

work page 2015
[10]

Hui Guan, Zhi-Jun Wei, Elizabeth Rumenova Rasolkova, and Chui- Jie Wu, Numerical simu- lations of two coaxial vortex rings head-on collision , Adv. Appl. Math. Mech 8 (2016), no. 4, 616–647

work page 2016
[11]

Stephen Gustafson, Evan Miller, and Tai-Peng Tsai, Regularity of axisymmetric, swirl-free Euler ﬂows in higher dimensions , preprint (2023)

work page 2023
[12]

O. A. Ladyˇ zenskaya, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry , Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 155–177. MR0241833

work page 1968
[13]

Lim and T.B

T.T. Lim and T.B. Nickels, Instability and reconnection in the head-on collision of tw o vortex rings, Nature 357 (1992), 225–227

work page 1992
[14]

Oshima, Head-on collision of two vortex rings , J

Y. Oshima, Head-on collision of two vortex rings , J. Phys. Soc. Japan 44 (1978), no. 44, 328– 331

work page 1978
[15]

Saint Raymond, Remarks on axisymmetric solutions of the incompressible Eu ler system , Comm

X. Saint Raymond, Remarks on axisymmetric solutions of the incompressible Eu ler system , Comm. Partial Diﬀerential Equations 19 (1994), no. 1-2, 321–334. MR1257007

work page 1994
[16]

Philippe Serfati, R´ egularit´ e stratiﬁ´ ee et ´ equation d’Euler3D ` a temps grand , C. R. Acad. Sci. Paris S´ er. I Math.318 (1994), no. 10, 925–928. MR1278153

work page 1994
[17]

Shariﬀ and A Leonard, Vortex rings, Ann

K. Shariﬀ and A Leonard, Vortex rings, Ann. Rev. Fluid Mech. 24 (1992), no. 24, 235–279

work page 1992
[18]

M. R. Ukhovskii and V. I. Yudovich, Axially symmetric ﬂows of ideal and viscous ﬂuids ﬁlling the whole space , J. Appl. Math. Mech. 32 (1968), 52–61. MR0239293 19

work page 1968

[1] [1]

Strain, Horng-Tzer Yau, and Tai- Peng Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equatio ns, Int

Chiun-Chuan Chen, Robert M. Strain, Horng-Tzer Yau, and Tai- Peng Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equatio ns, Int. Math. Res. Not. IMRN 9 (2008), Art. ID rnn016, 31. MR2429247

work page 2008

[2] [2]

arXiv:2210.07191

Jiajie Chen and Thomas Hou, Stable nearly self-similar blowup of the 2d boussinesq and 3 d euler equations with smooth data (2022). arXiv:2210.07191. 18

work page arXiv 2022

[3] [3]

Cheng, J

M. Cheng, J. Lou, and T.T. Lim, Numerical simulation of head-on collision of two coaxial vo rtex rings, Fluid Dyn. Res. 50 (2018), no. 24, 065513

work page 2018

[4] [4]

arXiv:2110.09079

Kyudong Choi and In-Jee Jeong, On vortex stretching for anti-parallel axisymmetric ﬂows (2021). arXiv:2110.09079

work page arXiv 2021

[5] [5]

C. Chu, C. Wang, C Chang, R Chang, and W Chang, Head-on collision of two coaxial vortex rings: Experiment and computation , J. Fluid Mech. 296 (1995), 39–71

work page 1995

[6] [6]

Nauk 62 (2007), no

Rapha¨ el Danchin, Axisymmetric incompressible ﬂows with bounded vorticity , Uspekhi Mat. Nauk 62 (2007), no. 3(375), 73–94. MR2355419

work page 2007

[7] [7]

, On perfect ﬂuids with bounded vorticity , C. R. Math. Acad. Sci. Paris 345 (2007), no. 7, 391–394. MR2361504

work page 2007

[8] [8]

Tarek Elgindi, Finite-time singularity formation for C1,α solutions to the incompressible Euler equations on R3, Ann. of Math. (2) 194 (2021), no. 3, 647–727. MR4334974

work page 2021

[9] [9]

Hao Feng and Vladim ´ ır ˇSver´ ak,On the Cauchy problem for axi-symmetric vortex rings , Arch. Ration. Mech. Anal. 215 (2015), no. 1, 89–123. MR3296145

work page 2015

[10] [10]

Hui Guan, Zhi-Jun Wei, Elizabeth Rumenova Rasolkova, and Chui- Jie Wu, Numerical simu- lations of two coaxial vortex rings head-on collision , Adv. Appl. Math. Mech 8 (2016), no. 4, 616–647

work page 2016

[11] [11]

Stephen Gustafson, Evan Miller, and Tai-Peng Tsai, Regularity of axisymmetric, swirl-free Euler ﬂows in higher dimensions , preprint (2023)

work page 2023

[12] [12]

O. A. Ladyˇ zenskaya, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry , Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 155–177. MR0241833

work page 1968

[13] [13]

Lim and T.B

T.T. Lim and T.B. Nickels, Instability and reconnection in the head-on collision of tw o vortex rings, Nature 357 (1992), 225–227

work page 1992

[14] [14]

Oshima, Head-on collision of two vortex rings , J

Y. Oshima, Head-on collision of two vortex rings , J. Phys. Soc. Japan 44 (1978), no. 44, 328– 331

work page 1978

[15] [15]

Saint Raymond, Remarks on axisymmetric solutions of the incompressible Eu ler system , Comm

X. Saint Raymond, Remarks on axisymmetric solutions of the incompressible Eu ler system , Comm. Partial Diﬀerential Equations 19 (1994), no. 1-2, 321–334. MR1257007

work page 1994

[16] [16]

Philippe Serfati, R´ egularit´ e stratiﬁ´ ee et ´ equation d’Euler3D ` a temps grand , C. R. Acad. Sci. Paris S´ er. I Math.318 (1994), no. 10, 925–928. MR1278153

work page 1994

[17] [17]

Shariﬀ and A Leonard, Vortex rings, Ann

K. Shariﬀ and A Leonard, Vortex rings, Ann. Rev. Fluid Mech. 24 (1992), no. 24, 235–279

work page 1992

[18] [18]

M. R. Ukhovskii and V. I. Yudovich, Axially symmetric ﬂows of ideal and viscous ﬂuids ﬁlling the whole space , J. Appl. Math. Mech. 32 (1968), 52–61. MR0239293 19

work page 1968