{"id":"13920837-c114-4d40-a982-011ff262981e","arxiv_id":"2607.10486","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Global weak martingale solutions exist for 2D/3D stochastic NSCHEs with transport noise under abstract assumptions, with pathwise uniqueness of strong solutions in 2D.","lead":"The paper proves global weak martingale solutions exist for the stochastic Navier-Stokes-Cahn-Hilliard system with transport noise in 2D/3D, plus pathwise uniqueness of strong solutions in 2D. It unifies two prior SPDE frameworks to handle this coupled diffuse-interface model.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged coercivity gap.","rationale":"The central claim (global weak martingale solutions under the abstract assumptions of Section 5, plus 2-D pathwise uniqueness) rests on a standard but indispensable spectral-gap condition between viscosity and transport-noise intensity. The reader already isolates this condition as the sole load-bearing restriction. My second-pass examination of the tightness argument on the non-metric space Z_T, the identification of the limit measure as a solution of the martingale problem, the recovery of the stochastic integral via the cylindrical Wiener process, and the 2-D uniqueness proof finds no further soft spot that would independently threaten the theorems. Consequently the CONDITIONAL verdict and MODERATE confidence remain appropriate; no adjustment is warranted.","tokens_in":102576,"tokens_out":523,"duration_ms":7072,"concrete_test":"Verify that the coercivity constant δ₀ appearing in (5.14)–(5.15) of Lemma 5.7 is strictly positive under the concrete transport coefficients of Assumption 3.16; if a physically relevant σ with δ₀=0 can be exhibited for which the Galerkin energy bound (6.10) still holds (or fails), the necessity of the spectral-gap condition is settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (coercivity (3.33)/(5.11) forcing transport-noise intensity strictly below viscosity) is correctly identified as load-bearing: without δ₀>0 the energy estimate (6.10) and subsequent Galerkin tightness collapse. After checking the hybrid Mikulevicius–Rozovskii / Brzeźniak–Motyl construction (tightness on Z_T, Prohorov-type weak convergence of laws, identification of the martingale problem via the stopped processes M^{n,z} and the representation (9.35)–(9.37)), the abstract assumptions of Section 5, the Landau-potential estimates, and the 2-D pathwise-uniqueness argument of Section 11, I find no additional internal inconsistency, circularity, or hidden gap that would independently undermine Theorem 5.10 or 5.11. The hybrid method is executed carefully and the estimates close under the stated hypotheses.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the stochastic Navier–Stokes–Cahn–Hilliard system (NSCHEs) with transport (gradient) noise on a bounded domain in R^d, d=2 or 3. Under abstract assumptions (coercivity of the noise relative to viscosity, linear growth of the multiplicative coefficients, Carathéodory regularity, Landau potential), the authors prove existence of a global weak martingale solution on any finite horizon T (Theorem 5.10) and, when d=2, pathwise uniqueness of the strong solution (Theorem 5.11 / 11.1). The argument proceeds by Galerkin approximation, uniform energy estimates, tightness of laws on a non-metric path space Z_T, weak convergence of measures via a Prohorov-type result, and identification of the limit as a solution of the martingale problem; the construction unifies the Mikulevicius–Rozovskii and Brzeźniak–Motyl frameworks.","tokens_in":102777,"tokens_out":919,"duration_ms":9532,"significance":"Transport noise is physically natural for binary fluid mixtures, yet previous stochastic NSCHE analyses treated only additive or non-gradient multiplicative noise. Establishing global weak martingale solutions in both 2D and 3D, together with 2D pathwise uniqueness, under a single abstract set of hypotheses is a genuine advance. The hybrid tightness–martingale-problem method is carefully executed and may be reusable for other coupled fluid–phase-field systems. The energy estimates, compactness criteria, and uniqueness argument are written out in detail and appear self-contained under the stated assumptions.","major_comments":[{"comment":"The coercivity condition (3.33)/(5.11) that forces the transport-noise intensity strictly below viscosity (δ₀>0) is load-bearing: without it the a-priori estimate (6.10) and all subsequent Galerkin tightness collapse. The paper correctly invokes the condition throughout Sections 5–6 and Lemma 5.7, but the physical range of admissible noise amplitudes is never quantified. A short remark on how restrictive (3.33) is for typical transport-noise models, or a pointer to literature where the spectral-gap condition is known to hold, would strengthen the applicability claim.","section":null},{"comment":"In the identification of the limit (Theorem 8.13 and Part 4 of the proof of Theorem 5.10), the representation of the martingale M via the cylindrical Wiener process (9.35)–(9.37) relies on Lemma B.8. The lemma is stated for the Gelfand triple (U,H,U'), yet the verification that the quadratic variation process satisfies the integrability needed for the lemma is only sketched via (9.20) and (8.46). A more explicit check that the stopped processes remain square-integrable under the Landau-potential growth would remove residual doubt about the passage from the local martingale problem to the Itô equation.","section":null}],"minor_comments":[{"comment":"The arXiv identifier in the header is 2607.10486; the year 2607 is almost certainly a typographical error and should be corrected before publication.","section":null},{"comment":"Notation for the chemical potential switches between μ, μ(ϕ) and μ_n without a uniform convention; a short glossary at the beginning of Section 2 would help the reader.","section":null},{"comment":"Several lengthy technical estimates are deferred to Appendices A–E. Cross-references in the main text (e.g., “by Lemma A.2”) are accurate, but a one-sentence roadmap of what each appendix contains would improve readability.","section":null},{"comment":"In the uniqueness section the process Y₂(t) defined in (11.4) is quite involved; a brief explanation of the origin of each term would make the Schmalfuss-trick argument easier to follow.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The hybrid method is solid and the paper is a natural fit for a probability/SPDEs journal. The only substantive limitation is the spectral-gap restriction on the noise, which is standard but should be flagged more clearly for applied readers. No concerns about novelty disclosure or citation pattern."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the first global weak-martingale existence result for the stochastic Navier-Stokes-Cahn-Hilliard system driven by genuine transport (gradient) noise, plus pathwise uniqueness in 2D. That is the concrete advance; earlier stochastic NSCHE work used additive or multiplicative noise of a different type.\n\nWhat they do well is the hybrid method. They take the Mikulevicius-Rozovskii martingale-problem route (no change of probability space once the space is rich enough) and marry it to the Brzeźniak-Motyl non-metric tightness machinery on the space Z_T. The Galerkin energy estimates (Props 6.1, 6.4, 6.8), the tightness argument (Lemma 7.14), the identification of the limit measure (Thm 8.13), and the 2-D uniqueness proof (Sec 11) are written out at the level of detail the field expects. The abstract framework in Section 5 is reusable for other coupled systems. Citations look honest; the only self-references are to the two methodological source papers that are actually used.\n\nThe soft spot is exactly the one the reader flagged: the spectral-gap coercivity (3.33)/(5.11) that forces the transport-noise intensity strictly below viscosity. Without δ_{0} > 0 the a-priori estimate (6.10) collapses and the whole argument stops. That condition is standard in the transport-noise literature, but it is load-bearing and physically restrictive. No other internal inconsistency or circularity turned up on a second pass through the hybrid construction and the appendices.\n\nThis is for people who already work on stochastic multiphase fluids or on abstract SPDE martingale problems. It is not a short paper and the technical overhead is high, but the architecture is sound. I would send it to referees; the result is new enough and the proofs careful enough to deserve a serious look, even if a referee will almost certainly ask whether the coercivity can be relaxed.","headline":"Solid first existence theory for NSCHEs with genuine transport noise, via a carefully executed hybrid of Mikulevicius-Rozovskii and Brzeźniak-Motyl; the coercivity gap is real but standard.","tokens_in":103403,"tokens_out":531,"would_cite":true,"duration_ms":11726,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","35Q30","35R60","76D05"],"pacs":[],"model":"grok-4.5","headline":"Transport-noise Navier–Stokes–Cahn–Hilliard mixtures admit global weak martingale solutions in 2D and 3D, with pathwise uniqueness in 2D.","keywords":["stochastic Navier–Stokes–Cahn–Hilliard","transport noise","weak martingale solutions","pathwise uniqueness","diffuse interface model","Galerkin approximation","martingale problem"],"falsifier":"Exhibit a smooth, divergence-free transport field for which the coercivity constant δ₀ vanishes and show that the corresponding Galerkin sequence loses its uniform energy bound, so that tightness fails.","tokens_in":103460,"feed_emoji":"🌊","tokens_out":625,"duration_ms":7499,"temperature":0.7,"pith_summary":"The paper studies a diffuse-interface model of two incompressible fluids whose velocity is driven by transport-type noise. It proves that, under a short list of abstract coercivity and growth conditions on the noise coefficients and with a regular Landau free-energy potential, global weak martingale solutions exist in both two and three space dimensions. In two dimensions the same solutions are pathwise unique (hence strong) once the multiplicative noise is Lipschitz. The argument unifies two previously separate approaches to stochastic PDEs: a Galerkin approximation that produces tight laws, followed by a direct identification of the limiting measure as a solution of a martingale problem on a non-metric path space, without changing the underlying probability space. The result supplies the first rigorous existence theory for this physically natural class of stochastic diffuse-interface models.","feed_headline":"Transport noise yields global weak solutions for fluid mixtures","feed_subtitle":"Existence in 2D/3D and uniqueness in 2D for stochastic Navier–Stokes–Cahn–Hilliard","key_machinery":"A Galerkin scheme whose laws are tight in a carefully chosen non-metric space Z_T of paths; the limiting measure is identified as a solution of the associated martingale problem by a generalization of Prohorov’s theorem that avoids the Jakubowski–Skorokhod representation.","core_discovery":"Under the abstract assumptions of Section 5 (coercivity of the transport noise relative to viscosity, linear growth of the multiplicative coefficients, Carathéodory regularity, and the Landau potential), the stochastic Navier–Stokes–Cahn–Hilliard system possesses at least one global weak martingale solution on any finite time horizon; when the spatial dimension is two the solution is pathwise unique.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Global weak solutions exist for 2D/3D stochastic fluid mixtures","Transport noise yields martingale solutions to stochastic NSCHEs","Pathwise uniqueness for 2D stochastic Navier-Stokes-Cahn-Hilliard","Unified framework proves global weak solutions for stochastic NSCHEs","Weak solutions to NSCHEs under transport noise in 2D and 3D"],"cache_read_input_tokens":98432,"weakest_assumption_plain":"The transport noise must be strictly weaker than the viscous dissipation (a spectral-gap condition that keeps the energy estimate closed).","fun_headline_variants_meta":{"raw":{"variants":["Global weak solutions exist for 2D/3D stochastic fluid mixtures","Transport noise yields martingale solutions to stochastic NSCHEs","Pathwise uniqueness for 2D stochastic Navier-Stokes-Cahn-Hilliard","Unified framework proves global weak solutions for stochastic NSCHEs","Weak solutions to NSCHEs under transport noise in 2D and 3D"]},"model":"grok-4.5","effort":"low","cost_usd":0.004228,"raw_usage":{"total_tokens":1158,"prompt_tokens":632,"num_sources_used":0,"completion_tokens":100,"cost_in_usd_ticks":42280000,"prompt_tokens_details":{"text_tokens":632,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":426,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":632,"tokens_out":100,"duration_ms":4150,"temperature":1.0,"reasoning_tokens":426,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T11:20:21.651336+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a smooth, divergence-free transport field for which the coercivity constant δ₀ vanishes and show that the corresponding Galerkin sequence loses its uniform energy bound, so that tightness fails.","supporting_citations":[],"review_version":1}