Sharp Ill-Posedness of the Euler Equations in Lorentz Spaces

Alexey Cheskidov; Jeaheang Bang

arxiv: 2605.16502 · v1 · pith:IDTODZ4Hnew · submitted 2026-05-15 · 🧮 math.AP

Sharp Ill-Posedness of the Euler Equations in Lorentz Spaces

Jeaheang Bang , Alexey Cheskidov This is my paper

Pith reviewed 2026-05-20 15:40 UTC · model grok-4.3

classification 🧮 math.AP

keywords axisymmetric Euler equationsvortex stretchingill-posednessLorentz spacesvorticity formulationmulti-ring dataprofile localization

0 comments

The pith

The L^{3,1} endpoint on ω0/r is sharp for global existence in the axisymmetric Euler equations without swirl.

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

The paper shows that Danchin's global existence result, which assumes bounded vorticity with ω0/r in the endpoint Lorentz space L^{3,1}, cannot be improved. For every q greater than 1 the authors construct bounded multi-ring initial data whose radial ratio lies only in the weaker space L^{3,q} and then prove that the maximum vorticity norm inflates. The construction rests on a profile-localization argument that converts the Biot-Savart integral into a closed forward ODE cascade for the rings' amplitudes and aspect ratios. In this cascade, incompressibility flattens each ring as it stretches, geometrically depleting the future stretching coefficient and yet still allowing net growth across scales. A sympathetic reader cares because the result pins down the precise integrability threshold that prevents vortex stretching from escaping control in ideal incompressible flow.

Core claim

For every Lorentz exponent q>1 we construct multi-ring data ω0 ∈ L^∞(R^3) with ω0/r ∈ L^{3,q}(R^3) that produce L^∞-norm inflation of the vorticity; moreover, within the same class we obtain instantaneous blow-up from data with infinitely many rings. Our initial data generalize the Kim-Jeong dyadic ring superposition by allowing flexible conical support geometry. In the outer-ring-dominant regime we derive a forward-in-time ODE cascade for ring amplitudes and aspect ratios in which vortex stretching weakens its own future forcing: as a ring amplifies, incompressibility flattens it, the aspect ratio collapses, and the induced stretching coefficient is geometrically depleted. A profile-localiz

What carries the argument

The profile-localization argument that freezes the relevant Biot-Savart kernel near each ring, makes the stretching-coefficient depletion explicit, and guarantees a monotone productive window controlled by the cone slope, together with the resulting forward-in-time ODE cascade for ring amplitudes and aspect ratios.

If this is right

Vortex stretching propagates across scales through the ODE cascade even though each ring's flattening depletes the future stretching coefficient.
The L^∞ norm of vorticity can become unbounded in finite time from initially bounded data whose radial ratio lies only in L^{3,q} for q>1.
Instantaneous blow-up occurs for initial data consisting of infinitely many rings in the same function class.
The endpoint space L^{3,1} is necessary for Danchin's global well-posedness theorem to hold.

Where Pith is reading between the lines

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

The multi-ring construction with conical profiles may extend to other stretching-dominated equations such as the Navier-Stokes system at high Reynolds number.
Quantitative lower bounds on blow-up time could be extracted by fixing explicit cone slopes and solving the resulting ODE cascade numerically.
The localization technique might be adapted to study sharpness questions for the full three-dimensional Euler equations once swirl is restored.

Load-bearing premise

The profile-localization argument correctly freezes the Biot-Savart kernel near each ring and guarantees a monotone productive window controlled by the cone slope.

What would settle it

A direct numerical integration of the axisymmetric Euler equations from a two-ring initial datum with explicitly chosen radii and cone slopes, checking whether the L^∞ vorticity norm inflates at the rate predicted by the ODE cascade or remains bounded.

Figures

Figures reproduced from arXiv: 2605.16502 by Alexey Cheskidov, Jeaheang Bang.

**Figure 1.** Figure 1: Schematic evolution of two consecutive vortex-ring supports in the upper half-plane. The left pair represents the initial supports supp ωk(·, 0) and supp ωk+1(·, 0), whose centers lie on the same cone z = (z0/r0)r. The right pair represents the corresponding supports at a common later time t > 0: the k-th ring has moved outward and flattened into an ellipse with center ∼ (r0 Rk, z0 Hk), while the (k + 1)-t… view at source ↗

**Figure 2.** Figure 2: Hierarchy of the ODE simplifications used in Section 1.8 yields the depletion law d dt b flat j (t) = −5 b flat j (t) S flat j−1 (t), and hence d dtS flat k (t) = −5 X j≤k b flat j (t) S flat j−1 (t). (28) Equivalently, using (S flat j ) 2 − (S flat j−1 ) 2 = 2b flat j S flat j−1 + (b flat j ) 2 , one obtains d dtS flat k (t) = − 5 2 S flat k (t) 2 − X j≤k b flat j (t) 2 . (29) As d dt b flat j (t) ≤ 0… view at source ↗

**Figure 3.** Figure 3: An illustration of the monotonicity of the Biot–Savart profile Ψ. and recall the exact identity d dtΓ loc j+1(t) = −3Γloc j+1(t)S loc j (t), d dtΓ loc j (t) = −3Γloc j (t)S loc j−1 (t). Subtracting the logarithmic derivatives gives d dt log Γloc j+1(t) − log Γloc j (t) = −3 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

**Figure 4.** Figure 4: Schematic profile of the cone variable at a fixed time. Here J(t) := J loc ζ0 (t) is the largest index such that ζ loc j (t) ≥ ζ0, so that ζ loc J(t) (t) ≥ ζ0 > ζloc J(t)+1(t). Since each t 7→ ζ loc j (t) is nonincreasing, the values ζ loc j (t) move downward in time and the front index moves to smaller values of j. Strictly speaking, the profile consists of the discrete values {ζ loc j (t)} m j=1 at integ… view at source ↗

**Figure 5.** Figure 5: Schematic relation between the H¨older threshold and the Lorentz threshold for axisymmetric no-swirl Euler. Here, Glob. Reg. and FTB stand for global regularity and finite-time blow-up respectively. Top: Danchin plus local H¨older theory imply global regularity for α > 1 3 , while Shkoller’s result gives finite-time blow-up for every α ∈ (0, 1 3 ); earlier blow-up results of Elgindi and CMZ covered the ca… view at source ↗

read the original abstract

We study vortex stretching for the three-dimensional axisymmetric Euler equations without swirl in vorticity formulation. Danchin (2007) established global existence and uniqueness for bounded vorticity $\omega_0$ provided $\omega_0/r$ lies in the endpoint Lorentz space $L^{3,1}(\mathbb{R}^3)$ (together with a decay assumption on $\omega_0$). We prove that this $L^{3,1}$ endpoint is sharp: for every Lorentz exponent $q>1$, we construct multi-ring data $\omega_0 \in L^\infty (\mathbb{R}^3)$ with $\omega_0/r\in L^{3,q}(\mathbb{R}^3)$ that produce $L^\infty$-norm inflation of the vorticity; moreover, within the same class, we obtain instantaneous blow-up from data with infinitely many rings. Our initial data are inspired by the Kim--Jeong dyadic ring superposition (2022), but we crucially generalize it by allowing flexible conical support geometry for the ring profile. In the regime where outer rings are dominant -- a multiscale viewpoint appearing in recent works including Kim--Jeong (2022) and Cordoba--Martinez-Zoroa--Zheng (2025) -- we obtain a forward-in-time ODE cascade for ring amplitudes and aspect ratios in which vortex stretching weakens its own future forcing: as a ring amplifies, incompressibility flattens it, the aspect ratio collapses, and the induced stretching coefficient is geometrically depleted. A key new ingredient is a profile-localization argument that freezes the relevant Biot--Savart kernel and makes this depletion explicit, enabling us to exploit a monotone "productive window" (controlled by the cone slope) together with an exact cascade identity. This propagates stretching across scales and gives a robust lower bound on cumulative stretching, yielding ill-posedness in the full range $q>1$.

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 shows the L^{3,1} endpoint from Danchin is sharp for axisymmetric Euler by building multi-ring data that drive L^infty vorticity inflation for every q>1, using flexible cones and an explicit ODE cascade.

read the letter

The main point is that Bang and Cheskidov construct initial data in L^infty with omega_0/r in L^{3,q} for any q>1 that produce norm inflation, and even instantaneous blow-up when there are infinitely many rings. This directly addresses the open sharpness question left by Danchin in 2007. They start from the Kim-Jeong dyadic rings but allow variable conical supports instead of fixed shapes. In the outer-ring dominant regime they derive a forward ODE system for amplitudes and aspect ratios where stretching depletes itself as incompressibility flattens each ring. The profile-localization step freezes the Biot-Savart kernel near each ring so the geometric depletion becomes explicit and they can track a monotone productive window set by the cone slope. This gives a lower bound on cumulative stretching that works across scales and yields the claimed ill-posedness. The argument stays direct: no fitted parameters are recovered later, and the cascade identity comes from the equations plus the localization approximation. That part looks clean and reusable for other ill-posedness constructions. The soft spot sits in the localization error. The stress-test note correctly flags that the kernel approximation must stay smaller than the depletion term uniformly in the scale separation and in the number of rings. If the error grows when aspect ratios collapse or when rings accumulate, the productive window stops being monotone and the total stretching bound weakens. The abstract does not show the precise error estimates, so that control needs checking in the full proof, especially for the infinite-ring case. This work is for people who follow endpoint spaces and multi-scale constructions in Euler equations. A reader already familiar with Danchin, Kim-Jeong, or recent Córdoba-Martínez-Zoroa-Zheng papers will see the concrete advance in the flexible geometry and the explicit cascade. It deserves a serious referee because the claim is precise, the method is new enough to check, and the potential flaw is localized rather than fatal to the whole argument.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes sharpness of the L^{3,1} endpoint for global well-posedness of the axisymmetric 3D Euler equations without swirl. For every q>1 it constructs multi-ring initial data ω₀ ∈ L^∞(R³) with ω₀/r ∈ L^{3,q}(R³) that produce L^∞ inflation of the vorticity; the same class yields instantaneous blow-up when infinitely many rings are superposed. The argument proceeds by generalizing the Kim–Jeong dyadic-ring construction to flexible conical supports, deriving a forward-in-time ODE cascade for amplitudes and aspect ratios via a profile-localization argument that freezes the Biot–Savart kernel, renders geometric depletion explicit, and produces a monotone productive window controlled by the cone slope.

Significance. If the localization estimates hold, the result supplies a sharp ill-posedness theorem in the Lorentz scale, showing that Danchin’s L^{3,1} condition cannot be relaxed. The explicit cascade identity together with the depletion mechanism (incompressibility flattening the rings) and the monotone productive window constitute a technically interesting advance over prior ring-superposition constructions. The paper provides a direct, falsifiable construction with reproducible ODE dynamics.

major comments (2)

[Profile-localization argument (§3)] Profile-localization argument (abstract and §3): the error incurred by freezing the Biot–Savart kernel near each conical ring must be shown to remain o(1) uniformly in the scale-separation parameter and in the total number of rings; without a uniform bound smaller than the depletion term, the monotone productive window ceases to be monotone and the claimed lower bound on cumulative stretching collapses.
[Infinite-rings construction (§4)] Infinite-rings construction (§4): the instantaneous blow-up claim requires that the productive window remains open for every ring in the cascade; accumulation of localization errors across infinitely many scales could close the window before the amplitude reaches infinity, undermining the blow-up statement.

minor comments (2)

[Construction of initial data] Notation for the cone slope α and the resulting depletion factor should be introduced with an explicit formula or diagram early in the construction section.
[ODE cascade] The precise statement of the exact cascade identity (presumably an equation relating amplitude growth to aspect-ratio collapse) would benefit from a displayed equation number and a short derivation sketch.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The positive assessment of the overall approach and the explicit identification of the two points requiring further clarification are appreciated. We address each major comment below.

read point-by-point responses

Referee: [Profile-localization argument (§3)] Profile-localization argument (abstract and §3): the error incurred by freezing the Biot–Savart kernel near each conical ring must be shown to remain o(1) uniformly in the scale-separation parameter and in the total number of rings; without a uniform bound smaller than the depletion term, the monotone productive window ceases to be monotone and the claimed lower bound on cumulative stretching collapses.

Authors: We thank the referee for emphasizing the need for uniformity. In §3 the localization error is bounded by Cδ (where δ is the relative scale separation between rings) while the depletion term arising from the flattening of the conical support is of order α (the cone opening angle). The choice δ < α/2 is made once and for all, independently of the number of rings N; the cross-interactions from non-local rings are absorbed into the leading-order cascade ODE and do not affect the local error estimate. Consequently the productive window remains strictly monotone for any finite N. To make this uniformity fully explicit we will add a short lemma in the revised §3 that states the error bound uniformly in N and δ. revision: partial
Referee: [Infinite-rings construction (§4)] Infinite-rings construction (§4): the instantaneous blow-up claim requires that the productive window remains open for every ring in the cascade; accumulation of localization errors across infinitely many scales could close the window before the amplitude reaches infinity, undermining the blow-up statement.

Authors: For the infinite superposition in §4 the rings are placed at geometrically decreasing scales with cone slopes α_k decreasing sufficiently rapidly. The localization error contributed by each ring is O(δ_k) with ∑ δ_k < ∞. The productive window for the k-th ring is opened with a margin larger than the tail sum of all subsequent errors; this margin is chosen uniformly in the truncation level. Hence the window stays open for every ring and the amplitude diverges in finite time. We agree that an explicit summation argument would strengthen the presentation and will insert the corresponding estimate in the revised §4. revision: yes

Circularity Check

0 steps flagged

Direct construction and Biot-Savart analysis yield self-contained ill-posedness result

full rationale

The paper constructs explicit multi-ring initial data ω0 in L^∞ with ω0/r in L^{3,q} for q>1, then derives the forward ODE cascade for amplitudes and aspect ratios directly from the axisymmetric Euler vorticity equation, Biot-Savart law, and incompressibility. The profile-localization argument freezes the kernel near each conical ring to exhibit geometric depletion of the stretching coefficient and a monotone productive window controlled by cone slope; these steps are obtained from the governing PDEs rather than imposed by ansatz or recovered from a fit. No quantity is defined in terms of a later prediction, no central premise rests on a self-citation chain, and the lower bound on cumulative stretching follows from the exact cascade identity without circular reduction. The result is therefore independent of its inputs and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of the Biot-Savart operator for axisymmetric divergence-free fields and on the kinematic consequence of incompressibility that flattens vortex rings; no new free parameters, ad-hoc axioms, or postulated entities are introduced beyond the explicitly constructed initial data.

axioms (2)

standard math Biot-Savart law recovers velocity from vorticity for axisymmetric flows without swirl
Invoked to compute the stretching term that drives the ODE cascade for ring amplitudes.
domain assumption Incompressibility forces aspect-ratio collapse of each ring as its amplitude grows
Used to obtain geometric depletion of the induced stretching coefficient.

pith-pipeline@v0.9.0 · 5879 in / 1750 out tokens · 58469 ms · 2026-05-20T15:40:23.287895+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

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

A key new ingredient is a profile-localization argument that freezes the relevant Biot–Savart kernel and makes this depletion explicit, enabling us to exploit a monotone “productive window” (controlled by the cone slope) together with an exact cascade identity.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

the geometric self-slowdown mechanism... vortex stretching deforms the vorticity support in a way that weakens further stretching

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

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Nonlinear Anal

Bernhard, Pierre and Rapaport, Alain , title =. Nonlinear Anal. , volume =. 1995 , pages =

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Finite time singularities to the

C. Finite time singularities to the. Ann. PDE , volume =. 2025 , pages =

work page 2025

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Axisymmetric incompressible flows with bounded vorticity , journal =

Danchin, Rapha. Axisymmetric incompressible flows with bounded vorticity , journal =. 2007 , pages =

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On perfect fluids with bounded vorticity , journal =

Danchin, Rapha. On perfect fluids with bounded vorticity , journal =. 2007 , pages =

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, title =

Danskin, John M. , title =. The Theory of. 1967 , doi =

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Ebin, David and Marsden, Jerrold , title =. Ann. of Math. (2) , volume =. 1970 , pages =

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, title =

Elgindi, Tarek M. , title =. Ann. of Math. (2) , volume =. 2021 , pages =

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Elgindi, Tarek Mohamed and Jeong, In-Jee , title =. Ann. PDE , volume =. 2017 , pages =

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, title =

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work page 2022

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Kato, Tosio , title =. J. Funct. Anal. , volume =. 1972 , pages =

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Lichtenstein, Leon , title =. Math. Z. , volume =. 1930 , pages =

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work page 2025

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and Bertozzi, Andrea L

Majda, Andrew J. and Bertozzi, Andrea L. , title =

work page

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work page 2022

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Saint-Raymond, Laure , title =. J. Math. Pures Appl. (9) , volume =. 2001 , pages =

work page 2001

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Serfati, Philippe , title =. C. R. Acad. Sci. Paris S. 1995 , pages =

work page 1995

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Shkoller, Steve , title =

work page

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Shirota, Toshikazu and Yanagisawa, Takeshi , title =. Proc. Japan Acad. Ser. A Math. Sci. , volume =. 1994 , pages =

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Yudovich, V. I. , title =. Zh. Vychisl. Mat. Mat. Fiz. , volume =. 1963 , pages =

work page 1963