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Invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations

T0 review · 2 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The stochastic one-dimensional Navier-Stokes-Korteweg equations admit invariant measures on a non-complete phase space.

desk verdict The paper gets invariant measures and the Feller property for 1D stochastic NSK by Krylov-Bogoliubov on a non-complete space, using domain choice plus damping to widen the admissible γ and α ranges beyond prior compressible-fluid work. read the letter →

arxiv 2606.07423 v1 pith:KXRSEAAP submitted 2026-06-05 math.AP

Donatella Donatelli , Lorenzo Pescatore , Stefano Spirito This is my paper · ORCID
classification math.AP
keywords invariantmeasuresstochasticNavier-Stokes-Kortewegone-dimensionalcompressiblefluidsKrylov-BogoliubovmethodFellerpropertycapillaritytensordampingterm
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves existence of invariant measures for a one-dimensional compressible viscous fluid with capillarity and density-dependent viscosity under stochastic additive noise. It applies the Krylov-Bogoliubov method despite the phase space being non-complete, by establishing that the Markov semigroup for strong solutions is Feller. This permits larger ranges of the adiabatic exponent gamma and viscosity exponent alpha than prior results for compressible fluids. A refined stability result on continuous dependence with respect to initial data is derived, and the role of the physical domain choice plus a damping term is highlighted.

What carries the argument

Krylov-Bogoliubov method applied to a Feller Markov semigroup on a non-complete phase space, enabled by the capillarity tensor and damping term.

What would settle it

An explicit construction showing that the Markov semigroup fails to be Feller for the stated ranges of gamma and alpha when the capillarity tensor is removed would disprove that the Korteweg term enables the result.

Watch

Extended reading notes

Core claim

We prove the existence of invariant measures by applying the Krylov-Bogoliubov method in a setting where the dynamics is supported on a non-complete phase space. This analysis is further enhanced by the derivation of a refined stability result determining the continuous dependence with respect to the initial data. The Markov semigroup associated with strong solutions is Feller and we can consider ranges of the adiabatic and viscosity exponents gamma and alpha larger than those available in the current ergodic literature for compressible fluids. The interplay between the choice of the physical domain and the use of a damping term is discussed.

Load-bearing premise

The specific choice of physical domain together with a damping term permits application of the Krylov-Bogoliubov method and the Feller property despite incompleteness of the phase space.

Editorial extensions

If this is right

  • The long-time behaviour of the fluid is described by the existence of invariant measures.
  • Strong solutions depend continuously on initial data.
  • The Feller property holds for the Markov semigroup associated with strong solutions.
  • Larger values of the exponents gamma and alpha are admissible compared to previous compressible fluid results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same approach may fail for these exponents without the capillarity tensor present.
  • Similar damping and domain choices could allow invariant measures in other one-dimensional stochastic fluid systems.
  • Uniqueness of the measures might hold under stronger assumptions on the noise strength or support.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript proves existence of invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations with additive noise by applying the Krylov-Bogoliubov method on a non-complete phase space, using the physical domain and a damping term. It establishes the Feller property of the Markov semigroup for strong solutions, derives a refined stability result on continuous dependence with respect to initial data, and obtains results for ranges of the exponents γ and α that exceed those in the existing ergodic literature for compressible fluids.

Significance. If the central claims hold, the work extends the ergodic theory of stochastic compressible fluids to include capillarity effects in one dimension, yielding Feller continuity and invariant measures under broader parameter regimes than previously available. The explicit discussion of the domain-damping interplay and the stability refinement provide concrete technical advances that could serve as a template for related stochastic PDE systems.

major comments (2)
  1. [Krylov-Bogoliubov application / invariant-measure existence section] The application of the Krylov-Bogoliubov theorem (presumably in the section deriving the invariant measure) relies on the claim that the specific physical domain together with the damping term circumvents the incompleteness of the phase space. The manuscript must explicitly construct an equivalent complete metric on the support of the dynamics or prove that every invariant measure is supported on a closed complete subset; without one of these steps the passage from tightness to existence is not guaranteed by the classical theorem.
  2. [Stability / Feller-property section] The refined stability result (continuous dependence on initial data) is invoked to obtain the Feller property. The proof should be checked for uniformity with respect to the noise intensity and the chosen ranges of γ and α; any hidden dependence on these parameters would restrict the claimed enlargement of the admissible exponent set.
minor comments (2)
  1. Notation for the capillarity tensor and the density-dependent viscosity should be introduced with a single consistent definition before the first appearance in the equations.
  2. The abstract states that results are obtained for ranges of γ and α larger than in the current literature; the introduction should contain an explicit comparison table or list of the previous admissible intervals versus the new ones.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important technical points regarding the application of the Krylov-Bogoliubov theorem and the uniformity of the stability estimates. We address each major comment below and indicate the revisions that will be made.

read point-by-point responses

  1. Referee: [Krylov-Bogoliubov application / invariant-measure existence section] The application of the Krylov-Bogoliubov theorem (presumably in the section deriving the invariant measure) relies on the claim that the specific physical domain together with the damping term circumvents the incompleteness of the phase space. The manuscript must explicitly construct an equivalent complete metric on the support of the dynamics or prove that every invariant measure is supported on a closed complete subset; without one of these steps the passage from tightness to existence is not guaranteed by the classical theorem.

    Authors: We agree that the classical statement of the Krylov-Bogoliubov theorem requires a complete metric space. In the manuscript the physical domain together with the damping term is used to guarantee that all trajectories remain in a bounded set on which the relevant norms are equivalent; however, we did not supply an explicit construction of a complete metric on the support nor a separate proof that every invariant measure is supported on a closed complete subset. In the revised version we will add a short subsection that either constructs an equivalent complete metric on the support of the dynamics or proves that the support of any invariant measure lies in a closed complete subset, thereby justifying the passage from tightness to existence. revision: yes

  2. Referee: [Stability / Feller-property section] The refined stability result (continuous dependence on initial data) is invoked to obtain the Feller property. The proof should be checked for uniformity with respect to the noise intensity and the chosen ranges of γ and α; any hidden dependence on these parameters would restrict the claimed enlargement of the admissible exponent set.

    Authors: The stability estimates are obtained by subtracting two strong solutions; the additive noise terms cancel, so the difference estimates are independent of the noise intensity. The admissible ranges for γ and α are determined by the integrability requirements in the a-priori bounds and the continuous-dependence argument; these requirements do not introduce additional restrictions beyond those already stated. In the revised manuscript we will insert a brief remark after the stability theorem explicitly confirming that all constants are uniform with respect to the noise intensity and that the exponent ranges remain unchanged. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: standard Krylov-Bogoliubov application with domain/damping adjustments on non-complete space

full rationale

The paper applies the Krylov-Bogoliubov method to establish existence of invariant measures for the stochastic Navier-Stokes-Korteweg system. The abstract explicitly notes that the dynamics live on a non-complete phase space and invokes the specific physical domain plus damping term to enable the argument, along with a Feller property for the Markov semigroup. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatz smuggling appear in the provided text. The derivation chain is presented as a direct application of the method with these adjustments; it does not reduce to its own inputs by construction. This is the expected honest non-finding for a self-contained theoretical proof.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The work rests on standard domain assumptions for the stochastic Navier-Stokes-Korteweg system and on the existence of strong solutions; no free parameters or invented entities are mentioned.

assumptions (2)
  • domain assumption The stochastic Navier-Stokes-Korteweg system with density-dependent viscosity and general capillarity admits strong solutions that generate a Markov semigroup.
    Required to apply Krylov-Bogoliubov and to discuss the Feller property.
  • domain assumption The phase space supporting the dynamics is non-complete, yet the Krylov-Bogoliubov method still applies when a damping term is present.
    Central technical premise stated in the abstract.

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Cite this review

Pith. "Pith review of Invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations." pith.science (2026). https://pith.science/paper/KXRSEAAP

@misc{pith2026260607423,
  author       = {Pith},
  title        = {Pith review of: Invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXRSEAAP}},
  note         = {Machine review of arXiv:2606.07423}
}
abstract

We investigate the long-time behaviour of a one-dimensional compressible viscous fluid with general capillarity and density dependent viscosity, driven by a stochastic additive noise. In particular, we prove the existence of invariant measures by applying the Krylov-Bogoliubov method in a setting where the dynamics is supported on a non-complete phase space. This analysis is further enhanced by the derivation of a refined stability result determining the continuous dependence with respect to the initial data. The present paper exhibits some properties and results for Korteweg fluids which are not known in absence of the capillarity tensor. In particular, we prove that the Markov semigroup associated with strong solutions is Feller and we can consider ranges of the adiabatic and viscosity exponents $\gamma$ and $\alpha$ larger than those available in the current ergodic literature for compressible fluids. Also the interplay between the choice of the physical domain and the use of a damping term is discussed.

Discussion (0). Continue with ORCID to comment.

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