{"id":"b31e6834-d4b5-47f8-be90-4766f4da3544","arxiv_id":"2606.07423","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence of invariant measures and Feller property of the Markov semigroup are established for 1D stochastic compressible Navier-Stokes equations with Korteweg capillarity.","lead":"The paper proves existence of invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations with capillarity and density-dependent viscosity via the Krylov-Bogoliubov method on a non-complete phase space. This extends ergodic analysis to fluid models with surface tension and permits wider ranges for the adiabatic and viscosity exponents.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Krylov-Bogoliubov on explicitly non-complete phase space requires extra justification beyond standard Polish-space arguments","rationale":"The reader's weakest_assumption isolates precisely the non-completeness issue; the proposed concrete_test directly checks whether the paper supplies the missing technical step that would make the application valid.","tokens_in":1701,"tokens_out":329,"duration_ms":12179,"concrete_test":"Locate the section defining the phase space X and the paragraph applying Krylov-Bogoliubov; verify whether X is shown to be Polish under an equivalent metric or whether the support of the tight family is proved to lie inside a complete closed subset of X. If neither holds, recompute the tightness argument on the metric completion of X and check whether the resulting measure is supported back in the original X.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim applies the Krylov-Bogoliubov theorem to obtain an invariant measure for the Markov semigroup on a phase space that the abstract states is non-complete. Standard statements of the theorem (and the supporting Prohorov theorem) presuppose a Polish space so that tightness implies relative compactness in the space of probability measures. The paper invokes the specific physical domain together with a damping term to circumvent this, yet the abstract supplies no indication that the authors either (i) exhibit an equivalent complete metric on the support of the dynamics or (ii) prove that all invariant measures are supported on a closed complete subset. Without one of these steps the passage from tightness to existence is not guaranteed by the classical theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1836,"tokens_out":518,"duration_ms":12543,"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":[{"comment":"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.","section":"Krylov-Bogoliubov application / invariant-measure existence section"},{"comment":"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.","section":"Stability / Feller-property section"}],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[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."},{"response":"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_made":"yes","referee_comment":"[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."}],"tokens_in":1344,"tokens_out":536,"duration_ms":11760,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is existence of invariant measures for the stochastic one-dimensional Navier-Stokes-Korteweg system, obtained via Krylov-Bogoliubov even though the phase space is not complete, together with the Feller property for the Markov semigroup on strong solutions and larger ranges for the exponents γ and α than appear in the existing ergodic literature for compressible fluids without capillarity.\n\nThe work does what it sets out to do on the new features that capillarity brings. The stability estimate for continuous dependence on initial data is a concrete technical step that supports the Feller claim. The discussion of how the physical domain and the damping term together keep the dynamics inside a workable set is a useful clarification that is not automatic in these models.\n\nThe soft spot is the non-complete space. Standard Krylov-Bogoliubov plus Prohorov requires a Polish space so that tightness yields relative compactness. The abstract states that the domain and damping circumvent the issue, but the proof must supply either an equivalent complete metric on the support or a demonstration that every invariant measure lives on a closed complete subset. If that step is carried out explicitly and without hidden assumptions, the argument is fine; if it is only sketched, the existence claim rests on an incomplete justification. The exponent ranges are presented as an improvement, but without the full comparison table it is hard to judge how much larger they really are.\n\nThis paper is for people already working on long-time behavior of stochastic compressible fluids or on Korteweg-type regularizations. A reader who needs the precise conditions under which invariant measures exist for these systems will get something usable from it. It is worth sending to a serious referee because the target is a genuine gap in the compressible stochastic literature and the method is standard enough that a referee can check the completeness fix in reasonable time.","headline":"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.","tokens_in":2309,"tokens_out":461,"would_cite":false,"duration_ms":13168,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The stochastic one-dimensional Navier-Stokes-Korteweg equations admit invariant measures on a non-complete phase space.","keywords":["invariant measures","stochastic Navier-Stokes-Korteweg","one-dimensional compressible fluids","Krylov-Bogoliubov method","Feller property","capillarity tensor","damping term"],"falsifier":"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.","tokens_in":2598,"feed_emoji":"","tokens_out":667,"duration_ms":17697,"temperature":0.7,"pith_summary":"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.","feed_headline":"Stochastic 1D Korteweg equations admit invariant measures","feed_subtitle":"Krylov-Bogoliubov applies despite incomplete phase space when capillarity and damping are used, extending gamma and alpha ranges.","key_machinery":"Krylov-Bogoliubov method applied to a Feller Markov semigroup on a non-complete phase space, enabled by the capillarity tensor and damping term.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["1D stochastic Korteweg fluids have invariant measures","Krylov-Bogoliubov on incomplete space yields invariants","Capillarity and damping enable Feller semigroup invariants","Extended exponent ranges for stochastic compressible invariants","Damping and domain interplay supports ergodic Korteweg results"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["1D stochastic Korteweg fluids have invariant measures","Krylov-Bogoliubov on incomplete space yields invariants","Capillarity and damping enable Feller semigroup invariants","Extended exponent ranges for stochastic compressible invariants","Damping and domain interplay supports ergodic Korteweg results","Invariant measures for stochastic Navier-Stokes-Korteweg in 1D"]},"model":"grok-4.3","cost_usd":0.004499,"raw_usage":{"total_tokens":2232,"prompt_tokens":650,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":44987000,"prompt_tokens_details":{"text_tokens":650,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1499,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":650,"tokens_out":83,"duration_ms":11114,"temperature":1.0,"reasoning_tokens":1499,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T21:22:04.880928+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}