{"id":"b8a4096f-10dd-47b7-a702-f0fc0fe75a15","arxiv_id":"2603.11762","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Global strong solutions exist for 2D/3D compressible Navier-Stokes-Korteweg on the torus with arbitrarily large data under BD viscosities and Bohm-type κ when ε ≤ ν.","lead":"The paper proves global-in-time strong solutions exist for the 2D and 3D compressible Navier-Stokes-Korteweg equations on the torus with arbitrarily large regular initial data, under a BD-type viscosity law and matching Bohm-type capillarity in the non-dispersive regime ε ≤ ν. This claims to close a long-open large-data existence question for capillary fluid models.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Abstract-only review leaves the claimed large-data a-priori estimates unverifiable; the BD/Bohm cancellations are necessary but their sufficiency for 3D global strong solutions cannot be checked.","rationale":"The Reader correctly flags that the BD/Bohm structural hypotheses are load-bearing and that an abstract-only review cannot certify soundness; the verdict UNVERDICTED with LOW confidence is therefore appropriate. My concern is essentially the same: the algebraic relations are necessary for the claimed cancellations, yet their sufficiency for closing the 3D large-data estimates remains an uncheckable assertion until the proofs appear. No independent evidence (machine-checked lemmas, publicly released code, or even a sketch of the key estimate) is supplied, so the gap cannot be closed by external means. Because the Reader already assigned UNVERDICTED, no adjustment of the verdict is warranted; the concrete test simply makes explicit the single verification that would settle the issue once the full text is released.","tokens_in":2199,"tokens_out":724,"duration_ms":6008,"concrete_test":"Once the full manuscript is available, extract the highest-order energy estimate (the one that controls ∥∇^{2}u∥_{L^{2}} and ∥∇^{3}\rho∥_{L^{2}} or their BD-effective-velocity analogues) and verify that every capillary contribution is absorbed by a viscous term under the sole assumption ε≤ν, with constants independent of the size of the initial data and without any smallness restriction on the vacuum set. If any residual term requires either ε≪ν or a positive lower bound on density, the large-data claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the precise BD algebraic relations μ(ρ)=ν\rho^α, λ(ρ)=2ν(α-1)\rho^α together with the generalized Bohm identity κ(ρ)=ε^{2}α^{2}\rho^{2α-3} and the non-dispersive restriction ε≤ν produce enough cancellations to close global-in-time strong a-priori estimates for arbitrarily large regular data on the 2D/3D torus. Because only the abstract is available, none of the energy estimates, higher-order estimates, or vacuum-control arguments can be inspected. The structural hypotheses are known to be necessary for the usual BD effective-velocity reformulation and for the cancellation of the most dangerous capillary terms, but necessity does not imply that the estimates actually close in three dimensions for large data; that is precisely the long-open step. Without the proofs one cannot confirm that the intermediary regime ε≤ν is used only for absorption rather than for a hidden smallness condition, nor that the vacuum set is controlled without additional restrictions on α. The load-bearing gap is therefore the unverifiable sufficiency of these cancellations for the 3D large-data problem.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript claims that the 2D and 3D compressible Navier-Stokes-Korteweg systems on the torus admit global-in-time strong solutions for arbitrarily large regular initial data, provided the viscosities satisfy the BD-type relations μ(ρ)=νρ^α and λ(ρ)=2ν(α-1)ρ^α and the Korteweg coefficient satisfies the generalized Bohm identity κ(ρ)=ε^{2}α^{2}ρ^{2α-3}, in the non-dispersive regime ε≤ν. The abstract presents this as the first such result for the 3D system under these structural hypotheses.","tokens_in":2418,"tokens_out":741,"duration_ms":5559,"significance":"If the claimed a-priori estimates close without hidden smallness, the result would resolve a longstanding open problem for the 3D compressible NSK system with large data in the non-dispersive regime. The structural hypotheses (BD viscosities plus Bohm-type capillarity) are standard devices that produce useful cancellations; a genuine large-data global strong existence theorem under precisely these relations would be a substantial contribution to the mathematical theory of capillary fluids. Because only the abstract is available, however, the technical novelty and the sufficiency of the cancellations cannot be verified.","major_comments":[{"comment":"The central claim of global strong solutions for arbitrarily large data in 3D rests on a-priori estimates that are not supplied. Without the energy hierarchy, higher-order estimates, vacuum-control arguments, or compactness passage, it is impossible to confirm that the BD/Bohm cancellations actually close the estimates or that the restriction ε≤ν is used only for absorption rather than as a hidden smallness condition. This is a load-bearing gap for the main theorem.","section":null},{"comment":"The abstract asserts that the result is the first for the 3D general NSK system with large data in the non-dispersive regime. In the absence of the proofs and of a precise comparison with existing literature (e.g., prior results under BD relations or dispersive regimes), the novelty claim cannot be audited and must be treated as provisional.","section":null},{"comment":"The range of the exponent α and the precise control of the vacuum set are not stated. For density-dependent viscosities of the form ρ^α, the admissible range of α is typically load-bearing for both the BD effective-velocity reformulation and the prevention of vacuum singularities; without this information the large-data claim remains incomplete.","section":null}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":"Only the abstract is available (arXiv:2603.11762). An abstract-only review cannot support a positive recommendation for a pure-existence claim whose entire content is the a-priori estimates. I recommend that the editor obtain the full manuscript (or wait for a complete submission) before soliciting a definitive report; the structural hypotheses are plausible but their sufficiency for 3D large-data global strong solutions is precisely the open step that must be checked line-by-line."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: the abstract asserts the first global-in-time strong solutions for the 3D compressible Navier-Stokes-Korteweg system (and 2D) with arbitrarily large regular data on the torus, under the BD viscosity relations μ=ν\rho^α, λ=2ν(α-1)\rho^α and the matching generalized Bohm capillarity κ=ε^{2}α^{2}\rho^{2α-3} with ε≤ν. That is the long-open large-data problem dating to Dunn–Serrin, and they frame it as solved in the non-dispersive regime.\n\nWhat is actually new is the 3D large-data claim itself. The tools (BD effective velocity, Bohm-type identities) are classical, so the contribution is closing the a-priori estimates without smallness. The abstract states the structural hypotheses cleanly and situates the result historically without overclaiming novelty of method. That is useful.\n\nThe soft spot is exactly the one the stress-test flags: we have only the abstract. The cancellations from those coefficient relations are necessary for the usual reformulation, but necessity does not guarantee the estimates close for large data in 3D, especially vacuum control and higher-order bounds. One cannot verify that ε≤ν is used only for absorption rather than a hidden smallness, or that no extra restriction on α sneaks in. That is a real gap for an abstract-only read; it is not a manufactured flaw. If the full proofs deliver the energy hierarchy as claimed, the central argument holds; right now it is unverifiable.\n\nThis paper is for mathematical fluid dynamicists working on capillary continuum models or BD systems. A reader who already knows the BD literature will get the most value and can audit the estimates once the full text appears. It deserves a serious referee: the claim is important enough within the field that an editor should send it out rather than desk-reject, even if heavy revision follows.\n\nRecommendation: accept for peer review. The result, if correct, is worth the referee time.","headline":"Claims first large-data global strong solutions for 3D NS-Korteweg under BD/Bohm in the non-dispersive regime, but only the abstract is here so the estimates stay unchecked.","tokens_in":3058,"tokens_out":596,"would_cite":false,"duration_ms":10527,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76N10","35B65"],"pacs":[],"model":"grok-4.5","headline":"The 2D and 3D compressible Navier-Stokes-Korteweg systems on the torus admit global-in-time strong solutions for arbitrarily large regular initial data when viscosities obey a BD-type power law and capillarity obeys a matching generalized B","keywords":["Navier-Stokes-Korteweg","global strong solutions","large initial data","BD viscosity","Bohm identity","capillarity","compressible fluids","non-dispersive regime"],"falsifier":"Construct a smooth, arbitrarily large initial datum on the three-dimensional torus for which the solution of the system with the stated BD-Bohm coefficients and ε≤ν loses regularity in finite time, or show that the a-priori estimates fail when the power-law exponents are perturbed by an arbitrarily small amount.","tokens_in":3071,"feed_emoji":"🌊","tokens_out":1007,"duration_ms":17397,"temperature":0.7,"pith_summary":"This paper proves that the compressible Navier-Stokes-Korteweg equations, which model capillary fluids, possess global strong solutions in both two and three space dimensions even when the initial density and velocity are arbitrarily large. The result requires that the shear and bulk viscosities satisfy the algebraic BD relation μ(ρ)=νρ^α and λ(ρ)=2ν(α−1)ρ^α, that the Korteweg coefficient is locked to the generalized Bohm form κ(ρ)=ε²α²ρ^{2α−3}, and that the capillarity constant does not exceed the viscosity constant. These structural hypotheses produce cancellations that close the a-priori estimates at the strong-solution level on the torus. The work thereby settles a longstanding open question, dating from Korteweg’s 1901 constitutive theory and the Dunn–Serrin formulation of 1985, for the non-dispersive regime and supplies the first such large-data existence theorem for the three-dimensional system under these constitutive laws.","feed_headline":"Global strong solutions for large-data 3D capillary fluids","feed_subtitle":"BD viscosities and a Bohm-type capillarity law with ε≤ν close the estimates on the torus","key_machinery":"The BD algebraic relation between the density-dependent viscosities, paired with the matching generalized Bohm form of the capillarity coefficient, produces exact cancellations that close the higher-order energy estimates and prevent finite-time blow-up of strong solutions for large data.","core_discovery":"Under the BD-type viscosity relations μ(ρ)=νρ^α, λ(ρ)=2ν(α−1)ρ^α and the generalized Bohm identity κ(ρ)=ε²α²ρ^{2α−3} with ε≤ν, the 2D and 3D compressible Navier-Stokes-Korteweg systems on the torus admit global-in-time strong solutions for any sufficiently regular initial data of arbitrary size.","pith_inferences":["The precise algebraic locking of viscosity and capillarity appears to be the mechanism that prevents finite-time singularity formation for capillary fluids under large data.","Analogous BD-Bohm pairings could be examined in other multiphase or phase-transition models that couple density-dependent viscosity with surface tension.","If the ratio ε/ν is allowed to exceed one, the dispersive regime may still admit large-data solutions under the same power laws, or may require an entirely different set of estimates.","Numerical schemes that exactly preserve the BD-Bohm identities may inherit unconditional stability for large-data simulations of capillary fluids."],"forward_implications":["Global strong solutions exist on the torus in two and three dimensions with no smallness restriction on the initial density or velocity.","Under the same constitutive laws the density stays positive and the solution remains regular for all positive times.","The intermediate non-dispersive regime ε≤ν, previously lacking large-data theory in three dimensions, is now covered.","The result supplies the first affirmative large-data existence theorem for the three-dimensional generalized Navier-Stokes-Korteweg system in the non-dispersive regime."],"fun_headline_variants":["Global strong solutions for large-data 2D/3D NS-Korteweg systems","Large-data global strong solutions via BD viscosities and Bohm capillarity","Global-in-time strong solutions for 3D compressible capillary fluids","Strong solutions for arbitrarily large data in 2D and 3D NS-Korteweg","BD-type laws yield global strong solutions for large-data capillary fluids"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The viscosities and the capillarity coefficient must obey the exact power-law identities that cancel in the energy estimates; if those algebraic relations are broken, the large-data a-priori bounds are not expected to close.","fun_headline_variants_meta":{"raw":{"variants":["Global strong solutions for large-data 2D/3D NS-Korteweg systems","Large-data global strong solutions via BD viscosities and Bohm capillarity","Global-in-time strong solutions for 3D compressible capillary fluids","Strong solutions for arbitrarily large data in 2D and 3D NS-Korteweg","BD-type laws yield global strong solutions for large-data capillary fluids"]},"model":"grok-4.5","effort":"low","cost_usd":0.007854,"raw_usage":{"total_tokens":1933,"prompt_tokens":888,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":78540000,"prompt_tokens_details":{"text_tokens":888,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":955,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":888,"tokens_out":90,"duration_ms":6625,"temperature":1.0,"reasoning_tokens":955,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T22:37:22.498524+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct a smooth, arbitrarily large initial datum on the three-dimensional torus for which the solution of the system with the stated BD-Bohm coefficients and ε≤ν loses regularity in finite time, or show that the a-priori estimates fail when the power-law exponents are perturbed by an arbitrarily small amount.","supporting_citations":[],"review_version":1}