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arxiv: 2604.21217 · v1 · submitted 2026-04-23 · 🧮 math.AP

Temporal decay estimates for global solutions of the Navier-Stokes equations with the Coriolis force

Tomoaki Yoshizawa This is my paper

Pith reviewed 2026-05-09 21:38 UTC · model grok-4.3

classification 🧮 math.AP
keywords Navier-Stokes equationsCoriolis forcetemporal decay estimatesglobal solutionsL^p normslinearized solutions
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The pith

Global solutions of the Navier-Stokes equations with Coriolis force decay as fast as their linearized versions when initial data is small.

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

This paper proves temporal decay estimates for global solutions to the Navier-Stokes equations that include the Coriolis force term. Under assumptions that include small initial data, the solutions are shown to decay in time at rates matching those of the corresponding linear equations. These rates exceed the decay expected from the heat equation alone. The estimates apply across all L^p norms for p ranging from 2 to infinity. Such results are useful for understanding the long-term behavior of rotating fluid flows.

Core claim

The paper shows that under several conditions including the smallness of the initial data, the solution decays as fast as the corresponding linearized solutions, and its decay rate is higher than expected from the flow of the heat equation. The estimates are derived for all L^p-norms with p∈[2, ∞].

What carries the argument

Decay estimates obtained by comparing the full nonlinear solutions to the linear system with the Coriolis term.

If this is right

  • The decay rates match the linear ones in every L^p norm from 2 to infinity.
  • The solutions decay faster than those of the heat equation.
  • The estimates hold for global-in-time solutions under the given conditions.

Where Pith is reading between the lines

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

  • The presence of the Coriolis force appears to improve the long-time decay properties compared to non-rotating cases.
  • These results may inform stability analysis in geophysical fluid models.
  • Similar techniques could be tested on other perturbations of the Navier-Stokes system.

Load-bearing premise

The smallness of the initial data together with other conditions that guarantee global existence of solutions.

What would settle it

Finding a small initial datum yielding a global solution whose L^p norm decays slower than the corresponding linear solution for some p between 2 and infinity.

read the original abstract

We consider temporal decay estimates for global solutions of the Navier-Stokes equations with the Coriolis force. We show that under several conditions including the smallness of the initial data, the solution decays as fast as the corresponding linearized solutions, and its decay rate is higher than expected from the flow of the heat equation. The estimates are derived for all $L^p$-norms with $p\in[2, \infty].$

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

3 major / 2 minor

Summary. The manuscript claims to establish temporal decay estimates for global solutions of the Navier-Stokes equations with the Coriolis force. Under several conditions including smallness of the initial data, the solutions are asserted to decay at the same rate as the corresponding linearized solutions, with this rate higher than that of the heat equation, and the estimates are given for all L^p norms with p ∈ [2, ∞].

Significance. If the claims hold with the necessary details supplied, the result would demonstrate that the Coriolis force can produce strictly improved temporal decay compared with the standard Navier-Stokes or heat equation setting. The uniform validity across the full range p ∈ [2, ∞] would strengthen applicability to pointwise and uniform estimates in rotating fluid models.

major comments (3)
  1. [Abstract] Abstract: The statement relies on 'several conditions including the smallness of the initial data' without specifying the precise functional space X in which smallness is measured or the global-existence theorem invoked. This is load-bearing, because the bootstrap for the Duhamel integral of the bilinear term must close in the same space that yields the improved linear decay for large p (including p = ∞).
  2. [Main results / linear estimates] The linear decay claim requires an explicit statement of the decay rates for the semigroup e^{t(A+C)} (A = Stokes, C = Coriolis) and a direct comparison showing strict improvement over the heat kernel rates; without these, the assertion that the nonlinear solution inherits the improved rate cannot be verified.
  3. [Nonlinear estimates / Duhamel term] The control of the nonlinear term via the mild formulation must be shown to preserve the linear decay rate without loss; the manuscript should supply the precise estimate on the bilinear operator that closes the a-priori bound for every p ∈ [2, ∞] under the stated smallness.
minor comments (2)
  1. [Abstract] The abstract would benefit from a concise statement of the main theorem, including the exact smallness norm and the functional setting.
  2. [Introduction] Notation for the Coriolis operator and the precise form of the linear operator should be introduced at the beginning of the introduction for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which will help improve the clarity of the presentation. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The statement relies on 'several conditions including the smallness of the initial data' without specifying the precise functional space X in which smallness is measured or the global-existence theorem invoked. This is load-bearing, because the bootstrap for the Duhamel integral of the bilinear term must close in the same space that yields the improved linear decay for large p (including p = ∞).

    Authors: We agree that greater precision in the abstract will aid readability. The smallness condition is imposed in the space X in which global existence is established (see Theorem 1.1), and the decay estimates are proved within the same framework to ensure the bootstrap closes. In the revised manuscript we will update the abstract to explicitly name this space X and reference the global-existence result, while retaining the concise style. revision: yes

  2. Referee: [Main results / linear estimates] The linear decay claim requires an explicit statement of the decay rates for the semigroup e^{t(A+C)} (A = Stokes, C = Coriolis) and a direct comparison showing strict improvement over the heat kernel rates; without these, the assertion that the nonlinear solution inherits the improved rate cannot be verified.

    Authors: The linear decay estimates for the semigroup e^{t(A+C)} are derived in Section 3 using the Fourier multiplier representation that incorporates the Coriolis term. These yield decay rates strictly faster than the corresponding heat-kernel rates for p > 2, owing to the additional oscillatory decay induced by the Coriolis force. We will add an explicit statement of these rates together with a side-by-side comparison to the heat-equation decay in a new paragraph at the beginning of Section 3, making the improvement transparent. revision: yes

  3. Referee: [Nonlinear estimates / Duhamel term] The control of the nonlinear term via the mild formulation must be shown to preserve the linear decay rate without loss; the manuscript should supply the precise estimate on the bilinear operator that closes the a-priori bound for every p ∈ [2, ∞] under the stated smallness.

    Authors: The mild formulation and the estimate for the bilinear term are treated in Section 4. The key bound is obtained by splitting the Duhamel integral and applying the linear decay estimates together with Hölder-type inequalities in the space X; smallness of the initial data in X ensures that the nonlinear contribution decays at the same rate as the linear part for all p ∈ [2, ∞]. To make this step fully explicit, we will insert a dedicated lemma stating the precise bilinear estimate and the resulting a-priori bound that closes the bootstrap uniformly in p. revision: yes

Circularity Check

0 steps flagged

No circularity: decay estimates derived from linear semigroup and Duhamel control under stated small-data assumptions

full rationale

The paper establishes temporal decay rates for global mild solutions of the Coriolis-Navier-Stokes system by comparing the nonlinear solution to the linear semigroup e^{t(A+C)} via the Duhamel formula. Under the smallness assumption in a suitable space that also guarantees global existence, the bilinear term is controlled so that the full solution inherits the linear decay rates (which are faster than the heat kernel for certain p). No step reduces a claimed prediction to a fitted parameter by construction, nor does any load-bearing uniqueness or ansatz rely on a self-citation chain that itself assumes the target decay. The derivation is self-contained once the linear estimates and small-data global existence are granted; the abstract's reference to 'several conditions' is an explicit hypothesis rather than a hidden tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard existence theory for small-data global solutions of Navier-Stokes and on linear decay estimates for the Coriolis-linearized system; no free parameters, ad-hoc axioms, or invented entities are visible from the abstract.

axioms (2)
  • domain assumption Global existence of mild solutions for sufficiently small initial data in appropriate function spaces
    Invoked to guarantee the solution exists for all time so that decay can be studied.
  • standard math Decay estimates for the linear Stokes-Coriolis operator are already known
    The paper compares nonlinear decay to the corresponding linearized decay.

pith-pipeline@v0.9.0 · 5355 in / 1291 out tokens · 30400 ms · 2026-05-09T21:38:04.944636+00:00 · methodology

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