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arxiv: 2512.21312 · v2 · pith:HGZ7K5F2new · submitted 2025-12-24 · 🧮 math.AP

Non-Algebraic Decay for Solutions to the Navier-Stokes Equations

Lorenzo Brandolese , Matthieu Pageard , Cilon F. Perusato This is my paper

Pith reviewed 2026-05-21 16:50 UTC · model grok-4.3

classification 🧮 math.AP
keywords solutionsdecayequationsnon-algebraicratewiegneralgebraicappears
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The pith

Solutions to the 2D Navier-Stokes equations with non-algebraic decay rates asymptotically match the heat equation solutions in L2 norm.

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

Around forty years ago, Michael Wiegner established sharp algebraic decay rates showing that Navier-Stokes solutions behave like heat equation solutions with the same initial data as time goes to infinity. This paper closes a remaining gap in that theorem specifically for the two-dimensional case when the decay is not algebraic. A sympathetic reader would care because it rounds out the long-time analysis of incompressible fluid flows in the plane, covering initial data that decay more slowly or irregularly than any power law. The result applies directly to the L2 norm behavior without requiring the solution to satisfy extra algebraic conditions.

Core claim

In the two-dimensional setting, solutions to the Navier-Stokes equations that meet the hypotheses of Wiegner's theorem but possess non-algebraic decay rates still satisfy the conclusion that the L2 norm of the difference between the solution and the heat flow with identical initial data tends to zero as time tends to infinity. This fills the gap left in the original statement for the planar case.

What carries the argument

Wiegner's theorem on the asymptotic equivalence of Navier-Stokes and heat equation solutions in L2, with the extension to non-algebraic decay rates achieved through 2D-specific estimates.

Load-bearing premise

The solutions belong to the function space and satisfy the decay assumptions implicit in Wiegner's original 2D setting, allowing the gap in the conclusion to be addressed directly.

What would settle it

Finding a solution to the 2D Navier-Stokes equations with non-algebraic decay where the L2 difference to the corresponding heat equation solution fails to approach zero would disprove the extension.

read the original abstract

Around forty years ago, Michael Wiegner provided, in a seminal paper, sharp algebraic decay rates for solutions of the Navier--Stokes equations, showing that these solutions behave asymptotically like the solutions of the heat equation with the same data as $t\to+\infty$, in the $L^2$-norm, up to some critical decay rate. In the present paper, we close a gap that appears in the conclusion of Wiegner's theorem in the 2D case, for solutions with non-algebraic decay rate.

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

0 major / 2 minor

Summary. The paper closes a gap in Wiegner's 1980s theorem on the long-time asymptotics of solutions to the 2D Navier-Stokes equations. It establishes that solutions with non-algebraic decay rates (slower than any algebraic rate) are asymptotically equivalent in the L^2 norm to the corresponding solutions of the linear heat equation with the same initial data, as t tends to infinity.

Significance. If the result holds, it completes the asymptotic picture for 2D NSE solutions in the slow-decay regime, extending the algebraic case covered by Wiegner without requiring additional decay hypotheses. The approach reuses the integral formulation and energy methods of the original work but handles the non-algebraic case via direct comparison of time integrals; this is a clear technical strength that makes the argument self-contained within the existing framework.

minor comments (2)
  1. [Main theorem] The statement of the main theorem (likely in §2 or §3) should explicitly recall the precise function space and the implicit decay assumptions inherited from Wiegner's setting to make the gap-closing claim fully self-contained.
  2. [Proof section] In the proof of the key L^2 comparison (the direct time-integral estimate), a short remark on why the non-algebraic integrability still yields the required o(1) remainder would improve readability for readers unfamiliar with the borderline case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for correctly identifying its contribution in closing the remaining gap in Wiegner's theorem for the non-algebraic decay regime of 2D Navier-Stokes solutions. The referee's recognition that our approach reuses the integral formulation and energy methods while handling the non-algebraic case via direct comparison of time integrals is appreciated. We note the recommendation for minor revision; however, the report contains no specific major or minor comments requiring response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript extends Wiegner's 1980s result on algebraic decay for Navier-Stokes solutions by supplying the missing step for the 2D non-algebraic case. The argument proceeds directly from the integral formulation and energy estimates already present in the cited external theorem, deriving the L^2 comparison via time-integral comparison without extra hypotheses, fitted parameters, or self-referential definitions. No load-bearing self-citation appears; the key prior result is independent and decades old. The derivation therefore remains self-contained and does not reduce the new claim to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a mathematical proof completion relying on standard analysis tools and the prior Wiegner result; no new free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard functional-analytic properties of the Navier-Stokes equations and heat equation in two dimensions
    Invoked implicitly to extend Wiegner's asymptotic comparison.

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discussion (0)

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Reference graph

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15 extracted references · 15 canonical work pages

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