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arxiv: 2605.15130 · v2 · pith:KYC67GB7new · submitted 2026-05-14 · 🧮 math.AP

Asymptotically Self-Similar Blowup for 3D Incompressible Euler with C^(1, 1/3-) Velocity II: 3D Profiles, Blowup, and Limiting behavior

Jiajie Chen This is my paper

Pith reviewed 2026-05-20 20:19 UTC · model grok-4.3

classification 🧮 math.AP
keywords 3D Euler equationsself-similar blowupvorticity profilessingularity formationaxisymmetric flowHolder regularityasymptotically self-similarfinite-time blowup
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The pith

Exact C^alpha self-similar blowup profiles for vorticity exist in the 3D Euler equations without swirl for every alpha below 1/3.

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 any alpha in the open interval from zero to one-third, the three-dimensional incompressible Euler equations without swirl admit exact self-similar blowup solutions whose vorticity belongs to the Holder class C^alpha. These profiles serve as building blocks to construct solutions that blow up in finite time from initial data whose vorticity is compactly supported and C^alpha while the velocity is C^{1,alpha} and square-integrable. The argument proceeds by lifting smooth one-dimensional blowup profiles into three dimensions through a fixed-point construction that handles the anisotropic structure with weighted estimates and repeated integration by parts along particle trajectories. As alpha approaches one-third from below the spatial blowup rate tends to infinity and the vorticity profile converges in a weighted supremum norm to a factorized form built from the limiting one-dimensional profile. This construction supplies a sharp example of singularity formation precisely at the regularity threshold where global regularity theorems cease to apply.

Core claim

For any alpha in (0, 1/3) exact C^alpha self-similar blowup profiles exist for the vorticity of the 3D incompressible Euler equation without swirl. From these profiles one obtains asymptotically self-similar blowup for initial data with compactly supported C^alpha vorticity and C^{1,alpha} intersect L^2 velocity. As alpha tends to one-third from below the spatial blowup rate diverges to infinity while the vorticity profile converges strongly in a weighted L^infty norm to a nonzero multiple of r to the power one-third times a smooth one-dimensional blowup profile.

What carries the argument

Fixed-point lifting of one-dimensional smooth blowup profiles to exact three-dimensional C^alpha profiles, implemented through anisotropic weighted estimates and double integration by parts along trajectories.

If this is right

  • Finite-time singularity formation occurs for the 3D Euler equations from an open set of initial data at every regularity below the known global-regularity threshold.
  • The blowup solutions become increasingly localized in space as alpha approaches one-third because the spatial rate diverges.
  • The vorticity profiles factorize and converge to a specific one-dimensional shape in the limit, giving a precise description of the critical blowup.
  • The same lifting technique yields stability of the profiles in a low-regularity topology, allowing construction of blowup from perturbed data.
  • Singularity formation in three dimensions can be obtained by transferring exact solutions from a one-dimensional nonlocal model.

Where Pith is reading between the lines

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

  • Similar lifting arguments may produce blowup examples for the axisymmetric Navier-Stokes equations at comparable regularity.
  • High-resolution numerical simulations near alpha equal to one-third could test whether the predicted divergence of the blowup rate and profile factorization appear in computed solutions.
  • The construction suggests that the critical regularity 1/3 separates regimes of global regularity from regimes permitting singularity formation for a wider class of axisymmetric flows.
  • The method of repeated integration by parts along trajectories may extend to other nonlocal transport equations that lack radial decay.

Load-bearing premise

The one-dimensional approximate profiles can be corrected into exact three-dimensional profiles by a fixed-point argument even though they lack sufficient decay in the radial direction.

What would settle it

An explicit computation or numerical solution of the profile equation demonstrating that no C^alpha fixed-point solution exists for some alpha strictly less than one-third would disprove the claimed existence of the blowup profiles.

Figures

Figures reproduced from arXiv: 2605.15130 by Jiajie Chen.

Figure 1
Figure 1. Figure 1: Lifting construction and limiting behavior as α → ( 1 3 ) −. Ω¯ α is the blowup profile for r −αω θ . W¯ α and Wα¯ are the blowup profiles for the 1D model, and ¯α = 1 3 . 1.2.1. Lifting 1D singularities. Following [43], we derive a gCLM-type 1D model (2.22) in z ∈ R along the axis r = 0. For α < 1 3 sufficiently close to 1 3 , we extend the blowup profiles W¯ α for the 1D model constructed in [11] constan… view at source ↗
Figure 1
Figure 1. Figure 1: Lifting construction and limiting behavior as α → ( 1 3 ) −. Ω¯ α is the blowup profile for r −αω θ . W¯ α and Wα¯ are the blowup profiles for the 1D model, and ¯α = 1 3 . We develop a family of anisotropic weighted estimates to capture the first two mechanisms, and prove Lemma 5.2 for the third ingredient. See Section 2.6 for the proof ideas for Theorem 1.1. Breakdown at α = 1 3 . The construction of 3D s… view at source ↗
read the original abstract

For any $\alpha \in (0,1/3)$, we construct exact $C^{\alpha}$ self-similar blowup profiles for the vorticity of the 3D incompressible Euler equation without swirl, and build on them to prove asymptotically self-similar blowup from $C_c^\alpha$ initial vorticity and $C^{1,\alpha}\cap L^2$ initial velocity. Moreover, we provide a complete characterization of the limiting behavior of the $C^{\alpha}$ vorticity profiles and the associated blowup solutions as $\alpha\to(1/3)^-$. Specifically, as $\alpha \to(1/3)^-$, the spatial blowup rate $\mathsf{c}_{\mathsf{x},\alpha}$ diverges to $\infty$, while the $C^{\alpha}$ vorticity profile $\Omega_{*,\alpha}^{\theta}$ asymptotically factorizes and converges strongly in a weighted $L^\infty$ norm to a nonzero constant multiple of $r^{1/3}\bar W_{1/3}(z)$, where $\bar W_{1/3}$ is a $C^\infty$ 1D blowup profile. Our construction is inspired by the Hou--Zhang blowup scenario. Using a fixed-point argument, we lift the $C^\infty$ blowup profiles for a 1D model constructed in the companion work [11] to exact 3D blowup profiles. To overcome the lack of $r$-directional decay in the approximate profile and capture the anisotropic structure, we develop a family of anisotropic weighted estimates and introduce a crucial integration-by-parts method along trajectories that exploits the equation twice. We then develop a finite codimension stability argument in a low-regularity setting to prove stability of the 3D profiles and establish asymptotically self-similar blowup. This blowup result is sharp in view of the global regularity theory for axisymmetric Euler without swirl with $ C_c^{\alpha}$ initial vorticity for all $\alpha \geq 1/3$. To the best of our knowledge, our results provide the first example in which a singularity from a 1D nonlocal fluid model is lifted to construct blowup for incompressible fluid equations in $\mathbb{R}^2$ or $\mathbb{R}^3$.

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 constructs exact C^α self-similar blowup profiles for the vorticity of the 3D incompressible Euler equation without swirl for any α ∈ (0,1/3) by lifting C^∞ 1D profiles from the companion paper [11] via a fixed-point argument. It then proves asymptotically self-similar blowup from C_c^α initial vorticity and C^{1,α} ∩ L² initial velocity, and characterizes the limiting behavior as α → (1/3)^−, showing that the spatial blowup rate c_{x,α} diverges to ∞ while the profile Ω_{*,α}^θ factorizes and converges strongly in a weighted L^∞ norm to a nonzero multiple of r^{1/3} W̄_{1/3}(z). The construction employs anisotropic weighted estimates and a double integration-by-parts along trajectories that exploits the vorticity equation twice.

Significance. If the fixed-point construction closes, the work supplies the first explicit lifting of a singularity from a 1D nonlocal model to an exact 3D Euler blowup profile, and the limiting analysis as α approaches the critical threshold 1/3 provides a sharp complement to existing global regularity theorems for α ≥ 1/3. The finite-codimension stability argument in the low-regularity setting is a technically substantial contribution.

major comments (2)
  1. [Fixed-point argument for 3D profiles] Fixed-point construction: The central claim that the double integration-by-parts along trajectories, combined with the family of anisotropic weighted estimates, absorbs all commutator and remainder terms in the C^α topology for α < 1/3 rests on the approximate profile lacking r-directional decay. Explicit control of the second-order remainders (generated when the flow map is only C^{1,α}) must be verified uniformly down to α = 0 and independent of the r-cutoff; without such bounds the contraction mapping may fail to close.
  2. [Limiting behavior as α → (1/3)^−] Limiting analysis: The strong convergence of Ω_{*,α}^θ to a multiple of r^{1/3} W̄_{1/3}(z) in the weighted L^∞ norm as α → (1/3)^−, together with the divergence of c_{x,α}, is load-bearing for the complete characterization. The passage to the limit inside the finite-codimension stability argument requires uniform estimates that remain valid when the spatial blowup rate becomes arbitrarily large.
minor comments (2)
  1. [Notation and function spaces] The definition of the anisotropic weights and the precise function spaces in which the fixed-point map is shown to be a contraction should be stated at the beginning of the construction section for immediate readability.
  2. [Estimates] A short table summarizing the dependence of all constants on α would help the reader track the uniformity of the estimates as α approaches 1/3.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of significance, and constructive major comments. We respond point-by-point below, indicating revisions where appropriate to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Fixed-point argument for 3D profiles] Fixed-point construction: The central claim that the double integration-by-parts along trajectories, combined with the family of anisotropic weighted estimates, absorbs all commutator and remainder terms in the C^α topology for α < 1/3 rests on the approximate profile lacking r-directional decay. Explicit control of the second-order remainders (generated when the flow map is only C^{1,α}) must be verified uniformly down to α = 0 and independent of the r-cutoff; without such bounds the contraction mapping may fail to close.

    Authors: We appreciate the referee's emphasis on this technical detail. The approximate profile indeed lacks r-decay by construction, as it is lifted directly from the 1D model. The anisotropic weights (Section 3) are chosen to grow in the r-direction while decaying rapidly in z, precisely to offset this. The double integration-by-parts along trajectories (detailed in the proof of Proposition 4.1) produces exact cancellation of the principal stretching terms from the vorticity equation, reducing the second-order remainders to integrals that are estimated via the C^{1,α} regularity of the flow map. These remainder bounds are derived in Lemmas 4.3–4.5 and are uniform for α ∈ (0,1/3), with constants that remain bounded as α → 0^+ and independent of the r-cutoff radius because the weights ensure sufficient decay outside a fixed compact set in z. We will add a short clarifying remark after Proposition 4.1 explicitly stating this uniformity and independence to make the closure of the contraction mapping fully transparent. revision: yes

  2. Referee: [Limiting behavior as α → (1/3)^−] Limiting analysis: The strong convergence of Ω_{*,α}^θ to a multiple of r^{1/3} W̄_{1/3}(z) in the weighted L^∞ norm as α → (1/3)^−, together with the divergence of c_{x,α}, is load-bearing for the complete characterization. The passage to the limit inside the finite-codimension stability argument requires uniform estimates that remain valid when the spatial blowup rate becomes arbitrarily large.

    Authors: We agree that uniformity in the limit is essential. The strong convergence of the profiles in the weighted L^∞ norm and the divergence of c_{x,α} are established in Theorem 6.1 via compactness in the weighted spaces together with the factorization property inherited from the 1D limiting profile. For the stability argument, the finite-codimension conditions are preserved under the limit because they are defined via continuous functionals on the weighted spaces. The estimates in Section 5 are obtained from the linearized operator around Ω_{*,α}, which converges to the operator around the limiting profile. To address the referee's concern directly, we will insert a new auxiliary result (Lemma 5.4) proving that the relevant operator norms and stability constants remain bounded uniformly for all sufficiently large c_x, using the scaling invariance of the Euler equations and the fact that the limiting profile satisfies the same 1D equation. This will be added in the revised manuscript. revision: yes

Circularity Check

1 steps flagged

Moderate circularity from self-citation load-bearing on companion 1D profiles for 3D lifting

specific steps
  1. self citation load bearing [Abstract]
    "Using a fixed-point argument, we lift the C^∞ blowup profiles for a 1D model constructed in the companion work [11] to exact 3D blowup profiles. To overcome the lack of r-directional decay in the approximate profile and capture the anisotropic structure, we develop a family of anisotropic weighted estimates and introduce a crucial integration-by-parts method along trajectories that exploits the equation twice."

    The existence of the exact 3D C^α self-similar vorticity profiles is obtained by lifting the 1D profiles whose construction is imported entirely from the self-cited companion paper [11]. While the lifting technique itself is new, the load-bearing premise (availability of the base 1D profiles) reduces to prior self-authored work without re-derivation or external verification in this manuscript.

full rationale

The paper's central construction of 3D C^α profiles and the subsequent stability/blowup results explicitly depend on lifting C^∞ 1D profiles constructed in companion work [11] (same authorship). This is a self-citation load-bearing step, but the fixed-point argument, anisotropic weighted estimates, double integration-by-parts along trajectories, and limiting analysis as α→(1/3)^- constitute substantial independent mathematical content. No self-definitional reductions, fitted inputs renamed as predictions, or ansatz smuggling are present. The derivation chain remains partially self-contained against external benchmarks once the 1D base is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of the 1D blowup profiles from the companion work and on standard functional-analytic tools for the fixed-point and stability arguments in anisotropic spaces.

axioms (1)
  • domain assumption The companion paper [11] constructs C^∞ 1D blowup profiles W̄_{1/3} for the reduced model.
    Invoked explicitly to lift the profiles to 3D via fixed-point.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Self-similar blow-up solutions of incompressible Euler equations in $\mathbb R^d, d\geq 3$ with $C^{1,1-2/d-}$ velocity

    math.AP 2026-05 unverdicted novelty 7.0

    Constructs self-similar blow-up solutions to axisymmetric Euler equations in d greater than or equal to 3 with initial data in C to the 1,alpha intersect smooth away from origin for alpha less than 1-2/d.

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