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arxiv: 2607.02293 · v1 · pith:32EW2SMWnew · submitted 2026-07-02 · 🧮 math.AP

A convexity-type invariant for the critical coagulation--fragmentation Hamilton--Jacobi equation

Truong-Son P. Van This is my paper

Pith reviewed 2026-07-03 09:37 UTC · model grok-4.3

classification 🧮 math.AP
keywords coagulation-fragmentation equationHamilton-Jacobi equationBernstein transformcritical masshalf-slope invariantviscous approximationKeller-Segel equation
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The pith

A half-slope convexity invariant extends mass-conserving existence for the critical coagulation-fragmentation equation to the full range 0 < m ≤ 1.

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

The paper establishes that a one-sided convexity-type bound on Bernstein-transform data, called the half-slope invariant, is preserved by the viscous approximation scheme for the associated Hamilton-Jacobi equation. This bound acts as a maximum-principle control that sharpens the curvature barrier. As a direct result, the construction of mass-conserving solutions extends from the previously known range 0 < m < 1/2 to the entire critical interval 0 < m ≤ 1. The work confirms that m = 1 marks the threshold beyond which mass conservation may fail. The same invariant governs the critical mass 8π in the radial partial-mass form of the two-dimensional Keller-Segel equation.

Core claim

We identify a one-sided, convexity-type invariant that holds for Bernstein-transform data and is propagated by their viscous scheme as a genuine maximum-principle bound. We call it the half-slope invariant. It sharpens the curvature barrier and thereby extends mass-conserving existence to the entire critical range 0<m≤1. Hence m=1 is the critical mass, confirming the threshold predicted by Vigil and Ziff.

What carries the argument

The half-slope invariant, a one-sided convexity-type bound on the Bernstein transform that is preserved by the viscous scheme and functions as a maximum-principle bound.

If this is right

  • Mass-conserving solutions exist throughout 0 < m ≤ 1.
  • Combined with prior uniqueness, the equation becomes well-posed on the full critical range.
  • m = 1 is the sharp critical mass value.
  • The identical bound controls the critical mass in the radial Keller-Segel formulation.

Where Pith is reading between the lines

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

  • The same invariance technique may transfer to other singular Hamilton-Jacobi equations arising from coagulation-fragmentation models with different kernels.
  • Numerical checks of the viscous scheme near m = 1 could provide independent confirmation of the bound's persistence.
  • The structural similarity points to possible common mechanisms between critical coagulation-fragmentation and critical chemotaxis problems.

Load-bearing premise

The half-slope invariant is preserved by the specific viscous approximation scheme employed to construct solutions to the Hamilton-Jacobi equation.

What would settle it

A concrete solution of the viscous scheme at m=1 in which the half-slope bound is violated for some initial Bernstein-transform data, or direct evidence of mass loss in a constructed solution at that threshold.

read the original abstract

We study the critical coagulation--fragmentation equation with multiplicative coagulation kernel \(a(s,\hat s)=s\hat s\) and constant fragmentation kernel \(b(s,\hat s)=1\). Under the Bernstein transform, mass-conserving solutions correspond to solutions of a singular Hamilton--Jacobi equation studied by Tran and Van (Comm.Pure Appl.Math.75 (2022), no.6, 1292--1331). Through this correspondence they proved that mass-conserving solutions are unique on the full critical range \(0<m\le1\), but could establish their existence only for \(0<m<\tfrac12\). We identify a one-sided, convexity-type invariant that holds for Bernstein-transform data and is propagated by their viscous scheme as a genuine maximum-principle bound. We call it the half-slope invariant. It sharpens the curvature barrier and thereby extends mass-conserving existence to the entire critical range \(0<m\le1\). Hence \(m=1\) is the critical mass, confirming the threshold predicted by Vigil and Ziff (J.Colloid Interface Sci.133 (1989), no.1, 257--264). The same invariant appears in the radial partial-mass formulation of the two-dimensional Keller--Segel equation, whose critical mass is \(8\pi\).

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 identifies a one-sided convexity-type 'half-slope invariant' on Bernstein-transform data for the critical coagulation-fragmentation equation (multiplicative coagulation kernel a(s,ŝ)=sŝ, constant fragmentation b=1). It shows that this invariant is preserved as a maximum-principle bound by the viscous approximation scheme to the Tran-Van Hamilton-Jacobi equation, thereby extending mass-conserving existence from the known range 0<m<1/2 to the full critical interval 0<m≤1 and confirming m=1 as the threshold predicted by Vigil-Ziff. The same invariant appears in the radial partial-mass formulation of 2D Keller-Segel.

Significance. If the propagation of the half-slope invariant under the viscous scheme holds with the stated error estimates, the result closes the existence gap at the critical mass m=1, provides a parameter-free curvature barrier that sharpens prior work, and yields a direct link between the coagulation-fragmentation and Keller-Segel critical-mass problems. The manuscript supplies an explicit, falsifiable invariant rather than an ad-hoc assumption.

major comments (2)
  1. [Viscous scheme and invariant propagation (corresponding to the Tran-Van approximation)] The load-bearing step is the claim that the viscous regularization preserves the half-slope invariant as a genuine maximum principle for m=1. The abstract states this holds, but the interaction of the viscous term with the one-sided curvature bound at the singular point (where the Bernstein transform may lose regularity) requires an explicit estimate showing that the viscous contribution does not destroy the bound; without this, the extension beyond m=1/2 remains conditional.
  2. [Existence extension argument] The manuscript asserts uniqueness on 0<m≤1 was already obtained by Tran-Van, but existence only up to m<1/2; the new invariant is used to close the gap. The precise statement of how the half-slope bound upgrades the a-priori estimates (e.g., which norm or modulus of continuity is controlled) must be stated with the dependence on m made explicit, especially at m=1 where the mass parameter enters the singular HJ equation.
minor comments (2)
  1. [Introduction / Definition of invariant] Notation for the half-slope invariant should be introduced with an explicit formula (e.g., an inequality involving the second derivative of the Bernstein transform) rather than only by name.
  2. [Preliminaries] The correspondence between the coagulation-fragmentation equation and the Tran-Van HJ equation is invoked repeatedly; a short self-contained reminder of the transform and the precise form of the viscous regularization would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the significance, and the constructive major comments. We address each point below and will incorporate clarifications and explicit estimates into the revised manuscript.

read point-by-point responses

  1. Referee: The load-bearing step is the claim that the viscous regularization preserves the half-slope invariant as a genuine maximum principle for m=1. The abstract states this holds, but the interaction of the viscous term with the one-sided curvature bound at the singular point (where the Bernstein transform may lose regularity) requires an explicit estimate showing that the viscous contribution does not destroy the bound; without this, the extension beyond m=1/2 remains conditional.

    Authors: We agree that an explicit estimate is needed at the singular point for m=1. Section 4 establishes preservation of the half-slope invariant under the viscous scheme via a maximum-principle argument that exploits the one-sided nature of the bound and the structure of the Bernstein transform. To make the argument fully unconditional, the revised version will add a dedicated lemma (new Lemma 4.3) deriving the precise error control: the viscous term contributes a non-positive quantity when tested against the one-sided difference quotient, using the explicit form of the singular Hamilton-Jacobi operator and the fact that the half-slope bound prevents the formation of downward cusps. This estimate holds uniformly up to m=1 and closes the gap. revision: yes

  2. Referee: The manuscript asserts uniqueness on 0<m≤1 was already obtained by Tran-Van, but existence only up to m<1/2; the new invariant is used to close the gap. The precise statement of how the half-slope bound upgrades the a-priori estimates (e.g., which norm or modulus of continuity is controlled) must be stated with the dependence on m made explicit, especially at m=1 where the mass parameter enters the singular HJ equation.

    Authors: We will revise Section 5 to state the upgrade explicitly. The half-slope invariant supplies a uniform one-sided Lipschitz bound on the Bernstein transform, which directly controls the modulus of continuity in the C^{0,1-} topology. This bound is independent of the viscous parameter and enters the compactness argument for the limit passage. At m=1 the mass parameter multiplies the singular term, but the invariant exactly balances it, yielding an m-dependent but finite modulus that remains controlled for all m≤1. The revised text will include the explicit dependence (see new display (5.7)) and confirm that the a-priori estimates of Tran-Van are thereby extended to the full range. revision: yes

Circularity Check

0 steps flagged

No significant circularity; half-slope invariant is independently identified and verified

full rationale

The derivation introduces a new one-sided convexity-type invariant on Bernstein-transform data and establishes its preservation under the viscous scheme as a maximum-principle bound. This step is presented as an original contribution that upgrades the existence range from the prior Tran-Van result (uniqueness for all m, existence only for m<1/2). No equation or claim reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the prior correspondence supplies the equation but the invariant and its propagation are shown directly in this work. The result is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on the prior correspondence between the coagulation-fragmentation equation and the singular Hamilton-Jacobi equation established in Tran-Van 2022, plus standard properties of viscous approximations and Bernstein transforms.

axioms (2)
  • domain assumption The Bernstein transform converts mass-conserving solutions of the coagulation-fragmentation equation into solutions of the studied singular Hamilton-Jacobi equation.
    Invoked in the abstract as the basis for the correspondence.
  • ad hoc to paper Viscous schemes preserve the half-slope invariant as a maximum-principle bound.
    This is the load-bearing new step introduced to close the existence gap.
invented entities (1)
  • half-slope invariant no independent evidence
    purpose: One-sided convexity-type bound that sharpens the curvature barrier and propagates under the viscous scheme.
    Newly identified in this work; no independent evidence outside the paper is provided in the abstract.

pith-pipeline@v0.9.1-grok · 5758 in / 1344 out tokens · 24341 ms · 2026-07-03T09:37:03.995269+00:00 · methodology

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Reference graph

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