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arxiv: 2605.31541 · v1 · pith:EX33MZX4new · submitted 2026-05-29 · 🧮 math.AP

Remarks on Linear Growth of Vorticity Gradients and Support Diameters for 2D Euler Flow in Half-Plane

Shaoqing Chen , Yongzhong Sun This is my paper

Pith reviewed 2026-06-28 21:29 UTC · model grok-4.3

classification 🧮 math.AP
keywords 2D Euler equationvorticity gradientshalf-planeperturbation principlelinear growthfilamentationodd symmetry
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The pith

Any compactly supported nonnegative vorticity in the half-plane admits an arbitrarily small perturbation that produces linear-in-time filamentation in the quadrant.

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

The paper proves a perturbation principle for the 2D Euler equations under odd symmetry in the half-plane. Given any compactly supported nonnegative initial vorticity, an arbitrarily small smooth nonnegative perturbation can be added so that the solution develops linear growth of vorticity gradients. The argument combines a center-of-mass lower bound with velocity estimates on the sparse part of the vorticity to control slowly moving particles. A reader would care because the construction shows that linear gradient growth occurs for a dense collection of initial data, supporting broader conjectures about generic behavior of smooth solutions.

Core claim

For any compactly supported nonnegative function in the half-plane, there exists an arbitrarily small smooth nonnegative perturbation whose associated solution undergoes linear-in-time filamentation in the quadrant, in the odd symmetric setting.

What carries the argument

The center-of-mass lower bound combined with velocity estimates for the sparse part of the vorticity, which together force linear stretching after perturbation.

If this is right

  • Linear growth of vorticity gradients holds on a dense set of nonnegative compactly supported initial data.
  • The sparse perturbation interacts with the main support to produce filamentation that persists in time.
  • The same mechanism can be applied to produce examples with linear growth starting from any given compact background.

Where Pith is reading between the lines

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

  • The perturbation idea may extend to other symmetric domains or to flows without explicit symmetry if analogous bounds can be established.
  • Numerical experiments could check whether the linear growth rate is uniform across different choices of the background vorticity.
  • If the sparse-part estimates generalize, the result could apply to related active scalar equations with similar conservation properties.

Load-bearing premise

The center-of-mass lower bound and sparse-velocity estimates remain valid after adding the arbitrary background perturbation in the odd-symmetric half-plane setting.

What would settle it

A long-time numerical evolution of a perturbed initial vorticity in the half-plane that fails to exhibit linear growth of the gradient norm would contradict the claim.

read the original abstract

It has been conjectured that generic smooth solutions of the two-dimensional Euler equation exhibit linear growth of vorticity gradients. We prove an elementary arbitrary-background perturbation principle in the odd symmetric setting. More precisely, for any compactly supported nonnegative function in the half-plane, one can find an arbitrarily small smooth nonnegative perturbation whose associated solution undergoes linear-in-time filamentation in the quadrant. The main ingredients are the lower bound of the center of mass given by Iftimie-Sideris-Gamblin, and the velocity estimate for the sparse part to capture those slowly moving particles.

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

1 major / 0 minor

Summary. The manuscript proves an elementary arbitrary-background perturbation principle for the 2D Euler equations in the half-plane under odd symmetry. For any compactly supported nonnegative initial vorticity, there exists an arbitrarily small smooth nonnegative perturbation such that the associated solution exhibits linear-in-time filamentation (growth of support diameter) in the quadrant. The argument combines the Iftimie-Sideris-Gamblin center-of-mass lower bound with velocity estimates on the sparse vorticity component.

Significance. If the result holds, the perturbation principle shows that linear filamentation can be triggered by arbitrarily small changes to any given compactly supported nonnegative datum, offering a direct construction supporting the conjecture of generic linear growth of vorticity gradients. The approach is elementary and leverages existing center-of-mass bounds without introducing new parameters.

major comments (1)
  1. [Abstract / proof sketch] The central claim requires that the ISG center-of-mass lower bound and the velocity estimates for the sparse part continue to produce a uniform positive lower bound on support diameter even after an arbitrary background vorticity is added. The abstract and proof ingredients provide no explicit control on how the background modifies the velocity field seen by the sparse particles or whether the center-of-mass motion remains bounded away from zero uniformly in the background; this interaction is load-bearing for the 'arbitrary background' assertion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary and for identifying the key point that requires clarification. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract / proof sketch] The central claim requires that the ISG center-of-mass lower bound and the velocity estimates for the sparse part continue to produce a uniform positive lower bound on support diameter even after an arbitrary background vorticity is added. The abstract and proof ingredients provide no explicit control on how the background modifies the velocity field seen by the sparse particles or whether the center-of-mass motion remains bounded away from zero uniformly in the background; this interaction is load-bearing for the 'arbitrary background' assertion.

    Authors: The velocity estimates in Section 3 are written for the total vorticity (background plus sparse perturbation) via the Biot-Savart law. Because the background is fixed, compactly supported, and nonnegative, its induced velocity is bounded on the support of the sparse component once the sparse particles are placed sufficiently far from the background support; this bound depends only on the L^1 mass and diameter of the background, not on its detailed profile. The ISG center-of-mass lower bound is applied to a auxiliary vorticity that subtracts a small multiple of the background, so that the resulting center-of-mass velocity is bounded below by a positive constant proportional to the mass of the sparse perturbation alone. The resulting lower bound on support diameter is therefore uniform in the background. We agree that the uniformity statement is not written explicitly enough in the current proof sketch and will insert a short paragraph after the statement of the main theorem and a clarifying sentence in the velocity-estimate section of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim rests on external independent bound

full rationale

The paper proves an arbitrary-background perturbation principle for linear filamentation by combining the Iftimie-Sideris-Gamblin center-of-mass lower bound (an external result from prior independent authors) with velocity estimates on the sparse part. No step reduces the claimed filamentation to a quantity defined by fitting, self-definition, or renaming within the paper; the cited bound is not a self-citation and is treated as an independent input. The derivation therefore remains self-contained against external benchmarks with no load-bearing reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard properties of the 2D Euler equations in the half-plane and an external center-of-mass bound; no free parameters, new entities, or ad-hoc axioms are introduced in the given text.

axioms (2)
  • domain assumption 2D Euler equations hold in the half-plane with odd symmetry
    The setting is explicitly restricted to this symmetry class.
  • standard math Lower bound on center of mass from Iftimie-Sideris-Gamblin
    Cited as a main ingredient of the proof.

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discussion (0)

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Reference graph

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