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arxiv: 2606.29512 · v1 · pith:TNWXCR6Onew · submitted 2026-06-28 · 🧮 math.PR · math-ph· math.AP· math.MP

On uniqueness of solutions to stochastic Navier--Stokes equations

Raymond Cotter , Istv\'an Gy\"ongy This is my paper

Pith reviewed 2026-06-30 02:06 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.APmath.MP
keywords stochastic Navier-Stokes equationsuniquenesscontinuous dependenceWiener processesPoisson martingale measuresgeneralization
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The pith

Theorems prove uniqueness and continuous dependence on initial conditions for stochastic Navier-Stokes equations driven by Wiener processes and Poisson martingale measures.

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

The paper presents theorems establishing that solutions to stochastic Navier-Stokes equations with driving noise from Wiener processes and Poisson martingale measures are unique. It also shows these solutions depend continuously on the initial condition. The results extend earlier theorems to this setting that combines continuous and jump noise. A sympathetic reader cares because uniqueness ensures the model produces a single well-defined trajectory for given starting data and random inputs, which is required for the equations to serve as a reliable description of random fluid motion.

Core claim

The central claim is that solutions to the stochastic Navier-Stokes equations driven by Wiener processes and Poisson martingale measures are unique and depend continuously on the initial condition. These theorems generalise some results from the cited work.

What carries the argument

Theorems on uniqueness and continuous dependence on the initial condition of solutions.

If this is right

  • The model produces a single probabilistic outcome for any fixed initial data and noise realisation.
  • Small changes in initial conditions produce only small changes in the solution paths.
  • The results apply to fluid models that include both continuous and discontinuous random forcing.

Where Pith is reading between the lines

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

  • Similar uniqueness arguments may apply to other stochastic evolution equations with mixed noise.
  • The continuous dependence property could be used to justify convergence of approximation schemes.

Load-bearing premise

The conditions needed to extend the earlier results to the combined Wiener and Poisson noise setting are satisfied.

What would settle it

An explicit pair of distinct solutions to the same equation with the same initial condition and the same driving noise would disprove uniqueness.

read the original abstract

Theorems on uniqueness and continuous dependence on the initial condition of solutions to stochastic Navier-Stokes equations driven by Wiener processes and Poisson martingale measures are presented. These theorems generalise some results from \cite{GK2026}.

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 / 0 minor

Summary. The manuscript presents theorems on uniqueness and continuous dependence on the initial condition for solutions to stochastic Navier-Stokes equations driven by Wiener processes and Poisson martingale measures. These results are stated to generalize some findings from the cited work GK2026.

Significance. If the proofs are correct, the results would extend the existing theory of stochastic Navier-Stokes equations to include jump noise via Poisson martingale measures. This is a natural and potentially useful generalization for modeling systems with discontinuous stochastic forcing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report correctly identifies the main contribution as theorems on uniqueness and continuous dependence for stochastic Navier-Stokes equations driven by both Wiener processes and Poisson martingale measures, extending results from GK2026.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states that it presents theorems on uniqueness and continuous dependence for stochastic Navier-Stokes equations driven by Wiener processes and Poisson martingale measures, explicitly as a generalization of results from the cited work GK2026. No derivation chain, equation, or central claim is shown to reduce by construction to a self-definition, fitted input renamed as prediction, or load-bearing self-citation whose validity depends on the present paper. The work consists of mathematical proofs under stated hypotheses, which are independent of the listed circularity patterns. The citation to prior results provides context for the generalization but does not substitute for or circularly justify the new theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract alone.

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Reference graph

Works this paper leans on

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