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arxiv: 2606.08352 · v1 · pith:54WI7TABnew · submitted 2026-06-06 · 🧮 math.AP · math-ph· math.MP

Finite-Scale One-Component Regularity via Harmonic Pressure for the 3D Navier-Stokes Equations

Runlong Yu This is my paper

Pith reviewed 2026-06-27 19:08 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP MSC 35Q30
keywords Navier-Stokes equationspartial regularityone-component regularityharmonic pressureCaffarelli-Kohn-Nirenberg criterionsuitable weak solutionsscale-invariant boundsfinite-scale decay
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The pith

Under a fixed scale-invariant bound on suitable weak solutions of 3D Navier-Stokes, smallness of the vertical velocity integral yields a positive lower bound on the local regularity radius at the origin.

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

The paper establishes an unconditional finite-scale one-component regularity result for the 3D incompressible Navier-Stokes equations. Given a suitable weak solution satisfying the scale-invariant local bound Phi(1) = A(1) + E(1) + C(1) + D(1) <= M, smallness of C_3(1) equal to the integral of |u_3|^3 over the unit cylinder produces a positive lower bound on the regularity radius at the origin that depends only on M. The proof converts the one-component smallness into approximation by a two-and-a-half-dimensional limiting class, then obtains Caffarelli-Kohn-Nirenberg smallness at a smaller scale by measuring pressure approximation in the quotient by spatially harmonic functions. Two further conditional layers supply logarithmic and power-type refinements under extra stability assumptions on the comparison class. The unconditional theorem is isolated from the conditional layers, which instead identify the quantitative stability needed to upgrade the compactness modulus.

Core claim

Under the fixed scale-invariant local bound Phi(1) <= M, smallness of the critical vertical-component quantity C_3(1) = integral over Q_1 of |u_3|^3 dx dt yields a positive lower bound, depending only on M, for the local regularity radius at the origin. The argument converts one-component smallness into approximation by the two-and-a-half-dimensional limiting class and then into Caffarelli-Kohn-Nirenberg smallness at a smaller scale, with the pressure approximation measured in a quotient by spatially harmonic functions. This pressure topology accounts for the obstruction that time-dependent harmonic pressures may have bounded scale-invariant L^{3/2}-oscillation while their pointwise gradient

What carries the argument

Conversion of one-component smallness into approximation by the two-and-a-half-dimensional limiting class, followed by pressure approximation measured in the quotient by spatially harmonic functions to reach Caffarelli-Kohn-Nirenberg smallness at a smaller scale.

If this is right

  • The unconditional theorem separates cleanly from the logarithmic and power-type conditional assumptions.
  • A prepared two-shadow comparison package replaces the abstract compactness modulus by a logarithmic modulus and produces logarithmic finite-scale decay.
  • Enlarging the comparison class to smooth no-stretching horizontal flows V=(v_h,0) with partial_3 pi allowed to be nonzero pairs the vertical residual with the small component u_3 in the relative-energy identity.
  • Under buffered strong-flow and localized relaxed stability inputs, the relaxed-shadowing layer yields power-type relaxed harmonic approximation and power-type finite-scale decay.
  • The logarithmic and power-type layers together identify the quantitative stability mechanisms required to upgrade the compactness modulus.

Where Pith is reading between the lines

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

  • The harmonic-pressure quotient may extend to other partial-regularity arguments that must handle time-dependent pressures with controlled oscillation but uncontrolled gradients.
  • If the conditional stability assumptions hold for typical numerical or physical approximations, the mechanism could yield explicit decay rates for regularity radii in simulations.
  • The separation of the unconditional result from the conditional layers suggests a modular strategy for strengthening one-component criteria in related fluid models.
  • Testing the conversion step in axisymmetric or two-and-a-half-dimensional reduced models would directly probe the two-and-a-half-dimensional limiting class used in the argument.

Load-bearing premise

The pressure approximation measured in the quotient by spatially harmonic functions converts into Caffarelli-Kohn-Nirenberg smallness at a smaller scale once the one-component smallness has been turned into approximation by the two-and-a-half-dimensional limiting class.

What would settle it

Existence of a suitable weak solution with Phi(1) <= M, C_3(1) arbitrarily small, yet with the local regularity radius at the origin equal to zero.

read the original abstract

We study a finite-scale one-component regularity mechanism for suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations. The results are organized in three layers. The first layer is unconditional. Under a fixed scale-invariant local bound Phi(1)=A(1)+E(1)+C(1)+D(1) <= M, smallness of the critical vertical-component quantity C_3(1)=int_{Q_1} |u_3|^3 dx dt yields a positive lower bound, depending only on M, for the local regularity radius at the origin. The proof converts one-component smallness into approximation by the two-and-a-half-dimensional limiting class and then into Caffarelli--Kohn--Nirenberg smallness at a smaller scale. The pressure approximation is measured in a quotient by spatially harmonic functions. This pressure topology reflects a genuine obstruction: time-dependent harmonic pressures may have bounded scale-invariant L^{3/2}-oscillation while their pointwise gradients lie beyond the control provided by the available scale-invariant quantities. The second layer is a conditional logarithmic refinement. A prepared two-shadow comparison package replaces the abstract compactness modulus by a logarithmic modulus and gives a logarithmic finite-scale decay. The third layer is a conditional relaxed-shadowing refinement. The comparison class is enlarged to smooth no-stretching horizontal flows V=(v_h,0), with the comparison pressure allowed to have partial_3 pi not equal to 0. The resulting vertical residual pairs with the small component u_3 in the relative-energy identity. Under the buffered strong-flow and localized relaxed stability inputs stated below, this gives a power-type relaxed harmonic approximation and a power-type finite-scale decay. The unconditional theorem is separated from the logarithmic and power-type assumptions; the latter two layers identify the quantitative stability mechanisms needed to upgrade the compactness modulus.

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 / 2 minor

Summary. The manuscript establishes a finite-scale one-component regularity mechanism for suitable weak solutions of the 3D incompressible Navier-Stokes equations. Under the scale-invariant bound Phi(1) = A(1) + E(1) + C(1) + D(1) <= M, smallness of the vertical-component quantity C_3(1) yields a positive lower bound (depending only on M) on the local regularity radius at the origin. The argument proceeds in three layers: an unconditional compactness reduction to a 2.5-dimensional limiting class followed by pressure control in a quotient by spatially harmonic functions to reach Caffarelli-Kohn-Nirenberg smallness; a conditional logarithmic refinement replacing the compactness modulus by a logarithmic modulus; and a conditional relaxed-shadowing refinement enlarging the comparison class to smooth no-stretching horizontal flows with a power-type decay under buffered strong-flow and localized relaxed stability assumptions. The unconditional result is separated from the conditional layers, which identify the quantitative stability mechanisms needed to upgrade the compactness modulus.

Significance. If the central claims hold, the work supplies a new partial-regularity criterion that isolates the role of a single velocity component and introduces a harmonic-pressure quotient topology designed to handle the obstruction that time-dependent harmonics can have controlled L^{3/2} oscillation yet uncontrolled gradients. The explicit separation of the unconditional theorem from the logarithmic and relaxed-shadowing layers, together with the identification of the compactness modulus as the sole non-quantitative ingredient, provides a clear roadmap for further quantitative improvements. The approach is internally consistent with the stated existence of an M-dependent positive radius and does not reduce the claimed lower bound to a fitted or self-referential parameter.

major comments (1)
  1. [Abstract, first layer] Abstract, first layer: the conversion from one-component smallness to 2.5D approximation and then to CKN smallness via the harmonic-pressure quotient is load-bearing for the unconditional claim that the regularity radius depends only on M. Explicit error estimates confirming that the quotient controls the required CKN quantities without introducing dependence outside M are needed to close the argument.
minor comments (2)
  1. [Abstract] The notation for the scale-invariant quantities A(1), E(1), C(1), D(1) that comprise Phi(1) is introduced without explicit definitions in the abstract; a brief parenthetical reminder of their standard meanings would improve readability.
  2. [Abstract] The manuscript flags the obstruction with time-dependent harmonics but does not indicate in the abstract where the full estimates verifying that the quotient circumvents this obstruction appear; a forward reference to the relevant section would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the identification of the load-bearing step in the unconditional layer. We address the major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract, first layer] Abstract, first layer: the conversion from one-component smallness to 2.5D approximation and then to CKN smallness via the harmonic-pressure quotient is load-bearing for the unconditional claim that the regularity radius depends only on M. Explicit error estimates confirming that the quotient controls the required CKN quantities without introducing dependence outside M are needed to close the argument.

    Authors: We agree that the argument requires explicit control on the error terms arising from the harmonic-pressure quotient. In the revised manuscript we will insert a new subsection (immediately following the compactness reduction) that derives quantitative bounds showing that the distance to the 2.5D class in the quotient topology produces an explicit smallness of the Caffarelli–Kohn–Nirenberg quantities at a smaller scale, with all constants depending only on the fixed scale-invariant bound M and the smallness parameter for C_3(1). These estimates close the unconditional claim without external dependence. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation chain begins from the given scale-invariant bound Phi(1) <= M together with smallness of the independent quantity C_3(1) and proceeds by compactness to a 2.5-D limiting class, followed by pressure approximation measured in the quotient by spatially harmonic functions to obtain CKN smallness at a smaller scale. None of these steps reduces the claimed positive lower bound on the regularity radius to a fitted parameter, a self-referential definition, or a self-citation chain; the pressure quotient is constructed precisely to handle the stated obstruction without importing the target conclusion. The logarithmic and relaxed-shadow layers are explicitly conditional on extra assumptions and do not affect the unconditional claim. The argument therefore remains self-contained against external mathematical tools (compactness, harmonic-function estimates, CKN theory) that are not defined in terms of the output radius.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the class of suitable weak solutions and on the existence of a pressure quotient topology that converts one-component smallness into CKN smallness; no free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • domain assumption The functions under study are suitable weak solutions of the three-dimensional incompressible Navier-Stokes equations.
    The abstract states that all results apply to this class.
invented entities (1)
  • harmonic pressure quotient topology no independent evidence
    purpose: To measure pressure approximation while allowing time-dependent harmonic pressures whose gradients escape scale-invariant control.
    Introduced in the first layer to handle the genuine obstruction noted in the abstract.

pith-pipeline@v0.9.1-grok · 5867 in / 1445 out tokens · 23339 ms · 2026-06-27T19:08:45.394635+00:00 · methodology

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