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arxiv: 2605.15149 · 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 I: C^{infty} 1D Limiting Profiles

Jiajie Chen
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:57 UTC · model grok-4.3

classification 🧮 math.AP
keywords incompressible Euler equationsself-similar blowupaxisymmetric flows1D modelsfixed-point constructionvorticity regularity
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The pith

For the critical case α=1/3 a C∞ self-similar blowup profile with unbounded stream function is constructed for a 1D model of 3D axisymmetric Euler.

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

The paper studies a one-parameter family of 1D models that capture the near-axis behavior of 3D axisymmetric incompressible Euler flow with C^α vorticity and no swirl. At the value α=1/3 the authors impose a normalization condition and produce a C∞ self-similar blowup profile whose 1D stream function is unbounded and whose spatial blowup rate is infinite. The construction proceeds by a fixed-point argument centered on a numerically computed approximate profile. For α slightly less than 1/3 the same profile is perturbed to obtain exact smooth profiles that instead have bounded stream function and finite blowup rate. These 1D objects are later used in a companion paper to build self-similar and asymptotically self-similar blowup solutions for the full 3D Euler system.

Core claim

At α=1/3, after imposing a normalization, there exists a C∞ self-similar blowup profile for the 1D model that possesses an unbounded 1D stream function and an infinite spatial blowup rate; the profile is obtained by a fixed-point argument around a numerically constructed approximate profile. For α<1/3 sufficiently close to 1/3, exact smooth 1D profiles with bounded stream function and finite spatial blowup rate are obtained by analytic perturbation of the α=1/3 profile.

What carries the argument

Fixed-point argument around a numerically constructed approximate profile, applied after a crucial normalization at α=1/3.

If this is right

  • Exact C∞ 1D self-similar blowup profiles exist at the critical regularity α=1/3.
  • For α slightly below 1/3 the profiles become bounded in the stream function with only finite spatial blowup rate.
  • These 1D profiles can be lifted to produce C^{1,α} self-similar blowup solutions for the 3D axisymmetric Euler equation without swirl.
  • Sharp asymptotically self-similar blowup holds for 3D axisymmetric Euler from C_c^α initial vorticity.

Where Pith is reading between the lines

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

  • The reliance on a numerical seed suggests that a fully rigorous error estimate for the approximate profile would turn the existence proof into a purely analytic statement.
  • The transition from infinite to finite blowup rate when α drops below 1/3 indicates a possible change in the leading-order singularity structure at the critical exponent.
  • If the 1D profiles survive the lifting procedure in the companion paper, they would give the first explicit examples of finite-time blowup in 3D Euler at the regularity threshold C^{1,1/3-}.

Load-bearing premise

The numerically constructed approximate profile lies sufficiently close to an exact solution that the fixed-point map is a contraction in the chosen function space.

What would settle it

An explicit bound on the approximation error of the numerical profile showing that the contraction constant exceeds one.

Figures

Figures reproduced from arXiv: 2605.15149 by Jiajie Chen.

Figure 1
Figure 1. Figure 1: Approximate Profiles for W¯ , ψ/x ˚ , ∂xψ˚ on grid points in [0, 100]. ψ˚ is the numerical evaluation of ψ˚(W¯ ) with superlinear growth. • Step 1: Update w(tn) → w(tn+1). Suppose that the coefficient ai(tn) at n-th step and at time tn for w(tn) has been obtained. To compute the forcing term F(w) for a given w in the representation (2.14) with coefficients ai , we use the formula (2.14b) and take the deriv… view at source ↗
Figure 2
Figure 2. Figure 2: Rigorous piecewise bounds over 8000 intervals covering [0, 1027]. Left: estimates relative to δF for the fixed-point map in Lemma 3.7, including linear, nonlinear, and error bounds associated with ϕ|W¯ |(BI + BII,1 + BII,3 + BII,2,III)(x), ϕ|W¯ |BN (x, δF )δF , ϕ|W¯ |δ −1 F BE (x, δF ). Right: estimate Bne(x) for contration estimate in Lemma 4.4. 4. Qualitative properties of F and a near-field contraction … view at source ↗
read the original abstract

We consider a one-parameter family of 1D models for the 3D axisymmetric incompressible Euler equation with $C^{\alpha}$ vorticity and without swirl near the symmetry axis. For $\alpha = \frac13$, we impose a crucial normalization and construct a $C^{\infty}$ self-similar blowup profile with unbounded 1D stream function and infinite spatial blowup rate, using a fixed-point argument around a numerically constructed approximate profile. For $\alpha < \frac13$ sufficiently close to $\frac13$, we perturb the $\frac13$-profile and analytically construct exact smooth 1D profiles with bounded stream function and finite spatial blowup rate. In the companion work~\cite{chen2026eulerII}, for any $\alpha \in (0,\frac13)$, we lift these 1D blowup profiles to construct exact $C^{1,\alpha}$ self-similar blowup profiles for 3D Euler, and build on them to prove sharp asymptotically self-similar blowup for 3D axisymmetric Euler without swirl from $C_c^\alpha$ initial vorticity and $C^{1,\alpha} \cap L^2$ initial velocity.

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 C^∞ self-similar blowup profiles for a one-parameter family of 1D models of the 3D axisymmetric incompressible Euler equations (with C^α vorticity, no swirl). For α=1/3, a normalization is imposed and a fixed-point argument is used around a numerically generated approximate profile to obtain an exact profile with unbounded 1D stream function and infinite spatial blowup rate. For α<1/3 sufficiently close to 1/3, this profile is perturbed to produce exact smooth profiles with bounded stream function and finite blowup rate. These 1D profiles are to be lifted to 3D C^{1,α} self-similar blowup solutions in the companion paper.

Significance. If the numerical seed is shown to lie inside the contraction ball with explicit residual bounds, the result supplies the first rigorous C^∞ self-similar profiles for this 1D Euler model at the critical α=1/3, together with a stable perturbation theory for nearby α. This is a concrete step toward asymptotically self-similar blowup constructions for 3D axisymmetric Euler from C_c^α data, and the fixed-point-plus-numerical-seed technique is a strength when the error control is complete.

major comments (2)
  1. [Section 3 (fixed-point setup and numerical seed)] The fixed-point construction for α=1/3 (centered on the numerically generated approximate profile) requires an explicit bound on the residual of that profile in the chosen Banach space (presumably a weighted C^∞ or Gevrey norm) together with a verified contraction constant; without these quantities the claim that the map is contractive on a ball containing the seed remains formally incomplete.
  2. [Section 4 (perturbation from the α=1/3 profile)] The perturbation argument for α<1/3 relies on the α=1/3 profile being C^∞ and satisfying the normalization; any gap in the error control for the seed profile propagates directly into the size of the perturbation interval and the claimed finite blowup rate.
minor comments (2)
  1. [Abstract and Section 3] The abstract states that the numerical profile is 'sufficiently close' but does not record the concrete norm or the size of the residual; this datum should appear explicitly in the main text or an appendix.
  2. [Section 2 (function spaces)] Notation for the weighted norms and the precise function space in which the contraction is proved should be introduced before the fixed-point map is defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We appreciate the positive assessment of the significance of the results and address the major comments point by point below. The requested explicit error controls will be added in the revision to complete the rigor of the fixed-point and perturbation arguments.

read point-by-point responses

  1. Referee: [Section 3 (fixed-point setup and numerical seed)] The fixed-point construction for α=1/3 (centered on the numerically generated approximate profile) requires an explicit bound on the residual of that profile in the chosen Banach space (presumably a weighted C^∞ or Gevrey norm) together with a verified contraction constant; without these quantities the claim that the map is contractive on a ball containing the seed remains formally incomplete.

    Authors: We agree that the argument is formally incomplete without explicit residual bounds and a verified contraction constant. The numerical seed is generated by a high-precision spectral method in a weighted Gevrey-type space that controls all derivatives. In the revised manuscript we will add a new subsection (or appendix) that computes an explicit upper bound on the residual norm of the seed using verified numerical integration (e.g., interval arithmetic on the quadrature errors) and shows that the Lipschitz constant of the fixed-point map is strictly less than 1 inside a ball of radius comparable to this residual. This closes the contraction-mapping argument and yields the exact C^∞ profile. revision: yes

  2. Referee: [Section 4 (perturbation from the α=1/3 profile)] The perturbation argument for α<1/3 relies on the α=1/3 profile being C^∞ and satisfying the normalization; any gap in the error control for the seed profile propagates directly into the size of the perturbation interval and the claimed finite blowup rate.

    Authors: We concur that quantitative control on the base profile is indispensable for the perturbation. Once the explicit residual bound and contraction constant are established for α=1/3, we will propagate these constants through the implicit-function theorem (or contraction-mapping perturbation) used in Section 4. The revision will include explicit formulas for the admissible interval of α below 1/3, together with bounds on the stream-function norm and the spatial blow-up rate that are uniform in that interval. This makes the finite-blow-up-rate statement fully rigorous. revision: yes

Circularity Check

0 steps flagged

Fixed-point construction around numerical seed is non-circular

full rationale

The paper establishes existence of the C^∞ 1D self-similar blowup profile for α=1/3 by a fixed-point argument centered on an independently generated numerical approximate solution. This numerical seed functions as external input data rather than a fitted parameter or self-defined quantity inside the derivation; the contraction mapping then produces the exact profile without reducing any claimed result to its own inputs by construction. No load-bearing self-citation, uniqueness theorem, or ansatz smuggling occurs in the core step, and the companion citation applies only to the subsequent 3D lifting. The derivation chain is therefore self-contained against external numerical benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a contraction mapping in a suitable Banach space once a sufficiently accurate numerical seed is supplied; no new physical entities are introduced.

free parameters (1)
  • normalization constant at α=1/3
    A scaling normalization is imposed to close the fixed-point argument; its specific value is chosen so that the approximate profile satisfies the required decay.
axioms (1)
  • standard math Banach fixed-point theorem applies in the chosen weighted Hölder space
    Invoked to guarantee a unique fixed point once the map is shown to be a contraction around the numerical seed.

pith-pipeline@v0.9.0 · 5528 in / 1378 out tokens · 34337 ms · 2026-05-15T02:57:11.234201+00:00 · methodology

discussion (0)

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Reference graph

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