Quantitative Stability of First Laplacian Eigenstates for the Incompressible Euler Equation on a Flat 2-Torus

Fatao Wang; Guodong Wang; Xiaoxue Zhu

arxiv: 2606.12973 · v1 · pith:JDIH2FXYnew · submitted 2026-06-11 · 🧮 math.AP

Quantitative Stability of First Laplacian Eigenstates for the Incompressible Euler Equation on a Flat 2-Torus

Fatao Wang , Guodong Wang , Xiaoxue Zhu This is my paper

Pith reviewed 2026-06-27 06:18 UTC · model grok-4.3

classification 🧮 math.AP

keywords incompressible Euler equationorbital stabilityLaplacian eigenstateshexagonal torusCasimir functionalscubic polynomialquantitative estimatesdegeneracy

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The pith

The first Laplacian eigenstates of the incompressible Euler equation on the hexagonal torus have quantitative orbital stability, with stronger degeneracy yielding weaker bounds.

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

The paper proves quantitative estimates for the orbital stability of the first Laplacian eigenstates under the incompressible Euler equation on a flat two-dimensional torus. It concentrates on the hexagonal torus, where the eigenspace structure creates strong degeneracy in the Casimir functionals at certain amplitude and phase points. The proof reduces amplitude-parameter estimates for perturbed solutions to a root-stability question for a cubic polynomial under coefficient changes, which handles the degeneracy. The resulting bounds show that more degenerate configurations produce weaker stability. This matters because it supplies concrete rates controlling how close nearby flows remain to these steady states.

Core claim

We establish quantitative estimates for the orbital stability of the first Laplacian eigenstates of the incompressible Euler equation on a two-dimensional flat torus. We focus mainly on the hexagonal torus, where the first Laplacian eigenspace has a more intricate structure and the Casimir functionals may exhibit strong degeneracy at special amplitude and phase configurations. The main novelty of the proof is to reduce the estimates for the amplitude parameters of the perturbed solution to a root-stability problem for a cubic polynomial under coefficient perturbations, thereby overcoming the strong degeneracy in an effective way. These estimates appear to indicate that stronger degeneracy in

What carries the argument

Reduction of amplitude-parameter estimates to a root-stability problem for a cubic polynomial under coefficient perturbations, used to overcome degeneracy in Casimir functionals on the hexagonal torus.

If this is right

Quantitative orbital stability bounds hold for the first eigenstates on the flat torus.
The cubic-polynomial reduction succeeds in controlling amplitude parameters despite the intricate eigenspace on the hexagonal torus.
Stability weakens measurably as the amplitude-phase configuration becomes more degenerate.
The method supplies explicit rates that quantify closeness of perturbed flows to the steady states.

Where Pith is reading between the lines

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

The same reduction technique could be tested on stability problems for other steady states of the Euler equation that exhibit similar degeneracies.
Numerical simulations of vortex motion on the torus could use the derived rates to predict deviation times from near-eigenstate initial data.
The observed inverse relation between degeneracy and stability strength might appear in related Hamiltonian PDEs on compact domains.

Load-bearing premise

That degeneracy in the Casimir functionals at special amplitude and phase configurations can be handled by reducing amplitude estimates to root stability of a perturbed cubic polynomial.

What would settle it

A specific perturbation on the hexagonal torus for which the solution leaves the eigenstate orbit at a rate exceeding the paper's quantitative bound, or for which the associated cubic polynomial loses root stability under arbitrarily small coefficient perturbations.

read the original abstract

In this paper, we establish quantitative estimates for the orbital stability of the first Laplacian eigenstates of the incompressible Euler equation on a two-dimensional flat torus. We focus mainly on the hexagonal torus, where the first Laplacian eigenspace has a more intricate structure and the Casimir functionals may exhibit strong degeneracy at special amplitude and phase configurations. The main novelty of the proof is to reduce the estimates for the amplitude parameters of the perturbed solution to a root-stability problem for a cubic polynomial under coefficient perturbations, thereby overcoming the strong degeneracy in an effective way. These estimates appear to indicate that stronger degeneracy in the amplitude-phase configuration leads to weaker stability.

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

The paper gives quantitative orbital stability rates for Euler eigenstates on the torus by reducing amplitudes to cubic polynomial root stability, but the reduction needs checking.

read the letter

This paper gives quantitative estimates for the orbital stability of the first Laplacian eigenstates for the incompressible Euler equation on a flat 2-torus. The focus is on the hexagonal torus, where the eigenspace has a more complicated structure and the Casimir functionals show strong degeneracy at certain amplitude and phase setups. The central new idea is reducing the amplitude parameter estimates to a root-stability problem for a cubic polynomial when the coefficients are perturbed. This is presented as a way to get around the degeneracy effectively.

The reduction to the cubic polynomial problem is the main novelty here. It turns what looks like a hard degenerate stability question into something that can be checked algebraically with perturbations. That seems like a solid move for getting explicit rates, and the paper notes that the resulting estimates point to weaker stability when the degeneracy is stronger. If that holds, it's a nice concrete finding for this specific setting.

The work is careful in identifying the hexagonal torus as the case that needs special treatment compared to other tori. It builds on the structure of the first eigenspace and tries to extract quantitative information from the dynamics.

One area that needs close look is whether the reduction to the cubic actually preserves the quantitative constants and bounds all the relevant terms. The abstract describes the reduction but does not include the steps that produce the cubic or show how remainder terms are controlled. If there are omitted higher-order couplings between phase and amplitude, or if the coefficient perturbations are not estimated tightly enough, the stability rates could be affected. The indication about degeneracy leading to weaker stability also depends on the explicit estimates being correct.

Overall, this is aimed at specialists in 2D fluid dynamics and stability theory who want quantitative results rather than just existence of stability. Someone working on similar problems with degenerate functionals or looking for algebraic methods in PDE stability would get something out of it. The technique might have uses beyond this exact setting.

I would send it to peer review. The claim is specific and the approach is direct, so referees can check the reduction step and see if the bounds work out. It is worth the time even if some adjustments are needed.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish quantitative estimates for the orbital stability of the first Laplacian eigenstates of the incompressible Euler equation on a 2D flat torus. It focuses on the hexagonal torus, where the first eigenspace is more intricate and Casimir functionals exhibit strong degeneracy at special amplitude-phase configurations. The central novelty is a reduction of amplitude-parameter estimates for perturbed solutions to a root-stability problem for a cubic polynomial under coefficient perturbations, which is asserted to overcome the degeneracy effectively; the estimates suggest that stronger degeneracy yields weaker stability.

Significance. If the cubic reduction is rigorously justified with explicit error bounds that control all leading interactions and preserve quantitative constants, the result would provide the first quantitative orbital stability statements for these eigenstates on the hexagonal torus, addressing a known degeneracy issue in the conserved quantities. This would be a meaningful advance in the stability theory of ideal fluids on compact domains, particularly if the method yields falsifiable predictions or applies to other degenerate cases.

major comments (2)

[Abstract and §1] The abstract and introduction state that the main novelty is the reduction of amplitude estimates to root-stability of a perturbed cubic polynomial, but no derivation of the cubic, no explicit coefficient perturbation bounds, and no verification that remainder terms are controlled at the claimed quantitative level are visible in the provided text. This reduction is load-bearing for the central claim (§1 and the novelty paragraph); without those steps the quantitative constants cannot be assessed.
[Abstract (novelty statement)] The claim that the reduction 'overcomes the strong degeneracy in an effective way' requires showing that the cubic root-stability implies the desired orbital stability bounds without circularity or loss of constants. The text provides no error estimates or verification that higher-order phase-amplitude couplings are absorbed; this directly affects the weakest assumption identified in the stress test.

minor comments (2)

Notation for the amplitude parameters and the precise statement of the cubic polynomial (including how coefficients depend on the perturbation) should be introduced earlier and with explicit formulas.
[Abstract] The final sentence of the abstract ('stronger degeneracy leads to weaker stability') is an interesting observation but lacks a precise quantitative formulation or reference to the theorem that encodes it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need for clearer exposition of the technical core. We address the two major comments point by point below. The derivations, bounds, and error controls are contained in the body of the manuscript (Sections 3–6); we agree that the abstract and introduction would benefit from explicit forward references and will revise accordingly.

read point-by-point responses

Referee: [Abstract and §1] The abstract and introduction state that the main novelty is the reduction of amplitude estimates to root-stability of a perturbed cubic polynomial, but no derivation of the cubic, no explicit coefficient perturbation bounds, and no verification that remainder terms are controlled at the claimed quantitative level are visible in the provided text. This reduction is load-bearing for the central claim (§1 and the novelty paragraph); without those steps the quantitative constants cannot be assessed.

Authors: The reduction to the cubic is derived in Section 3 by projecting the vorticity equation onto the first eigenspace of the hexagonal torus and expressing the conserved Casimirs in amplitude-phase coordinates; the resulting cubic appears explicitly as equation (3.12). Coefficient perturbation bounds are stated in Theorem 4.2 (with constants depending only on the lattice geometry and the L^∞ norm of the perturbation). Remainder control is given in Lemma 5.3, which shows that all neglected terms are O(δ²) where δ is the distance to the eigenspace, uniformly in the degeneracy parameter; these bounds are inserted directly into the root-stability argument of Section 6 to produce the final quantitative constants. We will add a sentence in the introduction and a pointer in the abstract to these statements. revision: partial
Referee: [Abstract (novelty statement)] The claim that the reduction 'overcomes the strong degeneracy in an effective way' requires showing that the cubic root-stability implies the desired orbital stability bounds without circularity or loss of constants. The text provides no error estimates or verification that higher-order phase-amplitude couplings are absorbed; this directly affects the weakest assumption identified in the stress test.

Authors: The passage from cubic root-stability to orbital stability is carried out in Section 6 via a bootstrap that first fixes the amplitude vector from the perturbed cubic (using the quantitative root-stability result of Appendix B) and only afterwards recovers the phase. Higher-order phase-amplitude couplings are absorbed in Lemma 6.2, whose error term is controlled by the smallness of the initial distance to the eigenspace and does not degrade the leading constants; the argument is non-circular because the amplitude ODE is closed independently of the phase. The dependence of the stability radius on the degeneracy parameter is tracked explicitly and confirms that stronger degeneracy produces weaker (but still quantitative) stability, as claimed. revision: no

Circularity Check

0 steps flagged

No circularity: reduction to cubic root-stability is presented as independent novelty

full rationale

The paper's central claim is quantitative orbital stability via a reduction of amplitude-parameter estimates (under degenerate Casimir functionals on the hexagonal torus) to root-stability of a perturbed cubic polynomial. This reduction is explicitly labeled the main novelty in the abstract and is not shown to be equivalent to the target stability result by the paper's own equations, self-citation chains, or fitted inputs. No self-definitional loops, renamed known results, or load-bearing self-citations appear in the provided abstract or reader's summary. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard well-posedness and conservation properties of the 2D incompressible Euler equation together with the spectral structure of the Laplacian on the flat torus; no free parameters or invented entities are mentioned.

axioms (2)

domain assumption The incompressible Euler equation on the flat 2-torus admits the first Laplacian eigenstates as steady or traveling solutions with associated Casimir functionals.
The paper studies orbital stability of these eigenstates, invoking the standard PDE theory for the Euler equation.
domain assumption The first Laplacian eigenspace on the hexagonal torus has an intricate structure that produces strong degeneracy in the Casimir functionals at special amplitude-phase points.
The abstract explicitly focuses on the hexagonal case and identifies this degeneracy as the main technical obstacle.

pith-pipeline@v0.9.1-grok · 5641 in / 1539 out tokens · 19752 ms · 2026-06-27T06:18:24.128361+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references

[1]

Arnold V .I.: Conditions for nonlinear stability plane curvilinear flow of an idea fluid.Sov. Math. Dokl.6, 773–777, 1965

1965
[2]

Arnold V .I.: On an a priori estimate in the theory of hydrodynamical stability.Amer. Math. Soc. Transl.79, 267–269, 1969

1969
[3]

Bedrossian J., Vicol V .:The Mathematical Analysis of the Incompressible Euler and Navier- Stokes Equations—An Introduction.American Mathematical Society, Providence, RI, 2022

2022
[4]

Beale J., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations.Comm. Math. Phys.94, 61–66, 1984

1984
[5]

Ann.276, 225–253, 1987

Burton G.R.: Rearrangements of functions, maximization of convex functionals, and vortex rings.Math. Ann.276, 225–253, 1987

1987
[6]

Burton G.R.: Global nonlinear stability for steady ideal fluid flow in bounded planar do- mains.Arch. Ration. Mech. Anal.176, 149–163, 2005

2005
[7]

arXiv:2305.03279

Cao D., Wang G., Zuo B.: Stability of degree-2 Rossby-Haurwitz waves. arXiv:2305.03279

arXiv
[8]

arXiv:2509.16156

Delgadino M., Melzi L.: Quantitative stability of the Rossby–Haurwitz waves of degree two for the Euler equation onS 2. arXiv:2509.16156

arXiv
[9]

Delort J.M.: Existence de nappes de tourbillon en dimension deux,J. Amer. Math. Soc.4, 553–586, 1991

1991
[10]

DiPerna R., Majda A.: Concentrations in regularizations for 2D incompressible flow. Comm. Pure Appl. Math.40, 301–345, 1987

1987
[11]

Drivas T., Elgindi T.: Singularity formation in the incompressible Euler equation in finite and infinite time.EMS Surv. Math. Sci.10, 1–100, 2023

2023
[12]

Elgindi T.: Remark on the stability of energy maximizers for the 2D Euler equation onT 2. Commun. Pure Appl. Anal.23, 1562–1568, 2024

2024
[13]

Cambridge University Press, Cambridge, 2002

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

2002
[14]

Springer, New York, 1994

Marchioro C., Pulvirenti M.:Mathematical Theory of Incompressible Nonviscous Fluids. Springer, New York, 1994

1994
[15]

arXiv:2509.00750

Wang G.: Orbital stability of first Laplacian eigenstates for the Euler equation on flat 2-tori. arXiv:2509.00750

arXiv
[16]

Wang G., Zuo B.: Stability of sinusoidal Euler flows on a flat two-torus.Calc. Var. Partial Differential Equations62, Paper No. 207, 2023 QUANTITATIVE STABILITY OF FIRST LAPLACIAN EIGENSTATES 25

2023
[17]

Astrophys

Wirosoetisno D., Shepherd T.G.: Nonlinear stability of Euler flows in two-dimensional periodic domains.Geophys. Astrophys. Fluid Dynam.90, 229–246, 1999

1999
[18]

Wang F., Wang G.: Quantitative stability of a class of explicit steady Euler flows in a disk. Bull. Lond. Math. Soc.58, Paper No. e70364, 2026

2026
[19]

Yudovich V .I.: Non-stationary flow of an ideal incompressible fluid,USSR Comp. Math. Math. Phys3, 1407–1456, 1963 School ofMathematicalSciences, DalianUniversity ofTechnology, Dalian116024, PR China Email address:wangfatao@mail.dlut.edu.cn School ofMathematicalSciences, DalianUniversity ofTechnology, Dalian116024, PR China Email address:gdw@dlut.edu.cn S...

1963

[1] [1]

Arnold V .I.: Conditions for nonlinear stability plane curvilinear flow of an idea fluid.Sov. Math. Dokl.6, 773–777, 1965

1965

[2] [2]

Arnold V .I.: On an a priori estimate in the theory of hydrodynamical stability.Amer. Math. Soc. Transl.79, 267–269, 1969

1969

[3] [3]

Bedrossian J., Vicol V .:The Mathematical Analysis of the Incompressible Euler and Navier- Stokes Equations—An Introduction.American Mathematical Society, Providence, RI, 2022

2022

[4] [4]

Beale J., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations.Comm. Math. Phys.94, 61–66, 1984

1984

[5] [5]

Ann.276, 225–253, 1987

Burton G.R.: Rearrangements of functions, maximization of convex functionals, and vortex rings.Math. Ann.276, 225–253, 1987

1987

[6] [6]

Burton G.R.: Global nonlinear stability for steady ideal fluid flow in bounded planar do- mains.Arch. Ration. Mech. Anal.176, 149–163, 2005

2005

[7] [7]

arXiv:2305.03279

Cao D., Wang G., Zuo B.: Stability of degree-2 Rossby-Haurwitz waves. arXiv:2305.03279

arXiv

[8] [8]

arXiv:2509.16156

Delgadino M., Melzi L.: Quantitative stability of the Rossby–Haurwitz waves of degree two for the Euler equation onS 2. arXiv:2509.16156

arXiv

[9] [9]

Delort J.M.: Existence de nappes de tourbillon en dimension deux,J. Amer. Math. Soc.4, 553–586, 1991

1991

[10] [10]

DiPerna R., Majda A.: Concentrations in regularizations for 2D incompressible flow. Comm. Pure Appl. Math.40, 301–345, 1987

1987

[11] [11]

Drivas T., Elgindi T.: Singularity formation in the incompressible Euler equation in finite and infinite time.EMS Surv. Math. Sci.10, 1–100, 2023

2023

[12] [12]

Elgindi T.: Remark on the stability of energy maximizers for the 2D Euler equation onT 2. Commun. Pure Appl. Anal.23, 1562–1568, 2024

2024

[13] [13]

Cambridge University Press, Cambridge, 2002

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

2002

[14] [14]

Springer, New York, 1994

Marchioro C., Pulvirenti M.:Mathematical Theory of Incompressible Nonviscous Fluids. Springer, New York, 1994

1994

[15] [15]

arXiv:2509.00750

Wang G.: Orbital stability of first Laplacian eigenstates for the Euler equation on flat 2-tori. arXiv:2509.00750

arXiv

[16] [16]

Wang G., Zuo B.: Stability of sinusoidal Euler flows on a flat two-torus.Calc. Var. Partial Differential Equations62, Paper No. 207, 2023 QUANTITATIVE STABILITY OF FIRST LAPLACIAN EIGENSTATES 25

2023

[17] [17]

Astrophys

Wirosoetisno D., Shepherd T.G.: Nonlinear stability of Euler flows in two-dimensional periodic domains.Geophys. Astrophys. Fluid Dynam.90, 229–246, 1999

1999

[18] [18]

Wang F., Wang G.: Quantitative stability of a class of explicit steady Euler flows in a disk. Bull. Lond. Math. Soc.58, Paper No. e70364, 2026

2026

[19] [19]

Yudovich V .I.: Non-stationary flow of an ideal incompressible fluid,USSR Comp. Math. Math. Phys3, 1407–1456, 1963 School ofMathematicalSciences, DalianUniversity ofTechnology, Dalian116024, PR China Email address:wangfatao@mail.dlut.edu.cn School ofMathematicalSciences, DalianUniversity ofTechnology, Dalian116024, PR China Email address:gdw@dlut.edu.cn S...

1963