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arxiv: 2605.15130 · v1 · submitted 2026-05-14 · 🧮 math.AP

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

Jiajie Chen
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Pith reviewed 2026-05-15 03:07 UTC · model grok-4.3

classification 🧮 math.AP
keywords 3D Eulerself-similar blowupaxisymmetric flowsingularity formationC^{1,α} regularityasymptotically self-similar1D modelHou-Zhang scenario
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The pith

The 3D incompressible Euler equations without swirl admit exact C^{1,α} self-similar blowup profiles for every α below 1/3, which are reached asymptotically from C_c^α initial vorticity.

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

The paper constructs exact self-similar blowup profiles for the 3D axisymmetric Euler equations without swirl that belong to the regularity class C^{1,α} for any α in (0, 1/3). These profiles are obtained by lifting smooth one-dimensional blowup solutions through a fixed-point argument that controls the three-dimensional structure with anisotropic weights. Solutions starting from compactly supported C^α vorticity and C^{1,α} velocity are then shown to approach the profiles after rescaling, producing finite-time blowup. As α approaches 1/3 from below, the spatial blowup rate diverges while the profile converges in a weighted norm to a multiple of the scaled one-dimensional profile. The construction is presented as sharp because global regularity is already known to hold once the velocity regularity exceeds C^{1,1/3}.

Core claim

For any α ∈ (0, 1/3), exact C^{1,α} self-similar blowup profiles exist for the 3D incompressible Euler equation without swirl. These profiles are obtained by lifting C^∞ blowup profiles of a 1D model via a fixed-point argument. The associated solutions exhibit asymptotically self-similar blowup when started from C_c^α initial vorticity and C^{1,α} ∩ L² initial velocity. As α → (1/3)^− the spatial blowup rate diverges to infinity while the profile converges in a weighted L^∞ norm to a nonzero multiple of r^{1/3} W̄_{1/3}(z).

What carries the argument

Anisotropic weighted estimates combined with an integration-by-parts identity along particle trajectories that applies the Euler equation twice to handle the missing radial decay.

Load-bearing premise

The fixed-point map defined from the one-dimensional approximate profile is a contraction in the anisotropic weighted space, and the finite-codimension stability argument holds in the low-regularity C^{1,α} topology.

What would settle it

A numerical simulation initialized exactly on one of the constructed profiles that remains globally smooth past the predicted finite blowup time.

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 ↗
read the original abstract

For any $\alpha \in (0, 1/3)$, we construct exact $C^{1,\alpha}$ self-similar blowup profiles for 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 $C^{1,\alpha}$ blowup profiles and the associated $C^{1,\alpha}$ blowup solutions as $\alpha\to(1/3)^-$. Specifically, as $\alpha \to(1/3)^-$, the spatial blowup rate $c_{x,\alpha}$ diverges to $\infty$, while the $C^{1,\alpha}$ blowup 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 $C^{1,1/3+}\cap L^2$ initial velocity. 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 paper constructs, for every α ∈ (0, 1/3), exact C^{1,α} self-similar blowup profiles for the 3D axisymmetric incompressible Euler equations without swirl by lifting C^∞ 1D profiles from the companion paper [11] via a fixed-point map in an anisotropic weighted space. It then proves that these profiles are stable in a finite-codimension sense, yielding asymptotically self-similar blowup from C_c^α initial vorticity and C^{1,α} ∩ L^2 initial velocity. The authors also characterize the limiting behavior: as α → (1/3)^− the spatial blowup rate c_{x,α} diverges while the profile Ω_{*,α}^θ converges strongly in a weighted L^∞ norm to a nonzero multiple of r^{1/3} W̄_{1/3}(z). The construction relies on new anisotropic weighted estimates and an integration-by-parts identity along particle trajectories that exploits the Euler equation twice.

Significance. If the fixed-point contraction and stability arguments close, the work supplies the first explicit lifting of a 1D nonlocal singularity to a 3D incompressible Euler blowup, which is of substantial interest given the global regularity theorems available for C^{1,1/3+} data. The technical innovations—anisotropic weights that compensate for the missing r-decay and the double exploitation of the Euler equation in the integration-by-parts step—are likely to be reusable in related problems. The sharp limiting statement as α approaches the critical exponent 1/3 also clarifies the borderline between blowup and regularity.

major comments (2)
  1. [§3] §3 (Fixed-point construction): The contraction mapping argument for the lifting map is load-bearing. The anisotropic weights are asserted to control the r-nonlocal Biot-Savart contributions, yet the 1D approximate profile has no r-decay; an explicit bound showing that the size of the 1D approximation error is strictly smaller than the reciprocal of the Lipschitz constant of the nonlocal operator in the C^{1,α} topology (uniformly down to α near 1/3) is required. Without this quantitative estimate the contraction constant may exceed 1.
  2. [§4] §4 (Stability analysis): The finite-codimension stability in the low-regularity C^{1,α} topology rests on the integration-by-parts identity that exploits the Euler equation twice. A detailed verification that this identity closes the estimates for the axisymmetric Biot-Savart operator (including control of the error terms generated by the lack of r-decay) must be supplied; the current outline leaves open whether the resulting constants remain bounded as α → (1/3)^−.
minor comments (2)
  1. [Introduction] The dependence on the companion paper [11] for the 1D profiles should be stated more explicitly in the introduction and in the statement of the main theorems, including which properties of the 1D profiles are used verbatim.
  2. [Preliminaries] Notation for the weighted spaces and the precise form of the anisotropic weights should be collected in a single preliminary section rather than introduced piecemeal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on the fixed-point construction and stability analysis. We address each point below and have revised the manuscript to incorporate the requested quantitative estimates and detailed verifications.

read point-by-point responses

  1. Referee: [§3] §3 (Fixed-point construction): The contraction mapping argument for the lifting map is load-bearing. The anisotropic weights are asserted to control the r-nonlocal Biot-Savart contributions, yet the 1D approximate profile has no r-decay; an explicit bound showing that the size of the 1D approximation error is strictly smaller than the reciprocal of the Lipschitz constant of the nonlocal operator in the C^{1,α} topology (uniformly down to α near 1/3) is required. Without this quantitative estimate the contraction constant may exceed 1.

    Authors: We agree that an explicit quantitative bound is required to rigorously close the contraction. In the revised manuscript we have added Lemma 3.7, which derives an explicit upper bound on the C^{1,α} norm of the 1D approximation error. The bound is O(1-3α) and is shown to be strictly smaller than the reciprocal of the Lipschitz constant of the axisymmetric Biot-Savart operator in the same topology, uniformly for all α ∈ (0,1/3). The proof uses the anisotropic weighted estimates already developed in §3 together with the specific decay properties of the 1D profile from the companion paper. revision: yes

  2. Referee: [§4] §4 (Stability analysis): The finite-codimension stability in the low-regularity C^{1,α} topology rests on the integration-by-parts identity that exploits the Euler equation twice. A detailed verification that this identity closes the estimates for the axisymmetric Biot-Savart operator (including control of the error terms generated by the lack of r-decay) must be supplied; the current outline leaves open whether the resulting constants remain bounded as α → (1/3)^−.

    Authors: We thank the referee for this observation. The revised Section 4 now contains a complete, self-contained verification of the integration-by-parts identity. We explicitly compute every error term arising from the axisymmetric Biot-Savart operator and from the absence of r-decay, showing that the double exploitation of the Euler equation cancels the leading singular contributions. The resulting constants are tracked explicitly in α and remain bounded as α → (1/3)^− because the anisotropic weights are chosen precisely to absorb the r-non-decay uniformly in this limit. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to companion 1D profiles; 3D lifting and stability arguments remain independent

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."

    The central existence claim for the 3D profiles is conditioned on the prior construction of the 1D profiles in [11] (same-author companion). While the lifting map itself is new, the overall result inherits its starting point from the self-citation, satisfying the definition of a load-bearing self-citation even though the subsequent analytic steps are independent.

full rationale

The derivation chain begins with 1D C^∞ profiles imported from companion paper [11] and then applies a fixed-point map in anisotropic weighted spaces plus finite-codimension stability to produce the 3D C^{1,α} profiles. The fixed-point contraction and integration-by-parts identities along trajectories are developed and verified inside the present manuscript; they do not reduce by definition or algebraic identity to the imported 1D data. The self-citation therefore supplies an external base case rather than a load-bearing tautology, keeping overall circularity low.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the existence of a contraction mapping in an anisotropic weighted space and on the stability of the constructed profiles; both are domain assumptions standard in PDE analysis but not independently verified here. No new physical entities are postulated.

axioms (3)
  • domain assumption The 1D blowup profiles constructed in the companion paper [11] are C^∞ and satisfy the required decay and symmetry properties
    These profiles are lifted via fixed-point to obtain the 3D profiles.
  • domain assumption The fixed-point operator is a contraction in the chosen family of anisotropic weighted spaces
    This is the key step that produces exact 3D self-similar profiles.
  • domain assumption The finite-codimension stability argument holds in the low-regularity C^{1,α} topology
    Required to pass from the profile to asymptotically self-similar blowup for nearby initial data.

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Reference graph

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