{"id":"4e4960a6-7295-47d9-b5ec-fa336b594e4b","arxiv_id":"2604.23652","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Global strong solutions to the compressible NSK system exist in R^2 and R^3 for arbitrarily large initial data with non-vacuum far-field density.","lead":"This paper proves global-in-time strong solutions exist for the multi-dimensional compressible Navier-Stokes-Korteweg system on the whole space R^N (N=2,3) with arbitrarily large initial data and positive far-field density. A smart generalist might read it to see how mathematical fluid models are extended from confined periodic boxes to open unbounded domains that better match real physical settings.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Modified Nash-Moser iteration's control of low-frequency terms in R^N may require implicit decay not guaranteed by arbitrary large data with only non-vacuum far-field.","rationale":"The reader's weakest assumption correctly flags the iteration scheme as the load-bearing technical step. The concern is not the algebraic relations on coefficients (which are explicit hypotheses) but whether the whole-space adaptation truly closes without hidden decay. This is a concrete internal-consistency question rather than an external-consensus issue. If the low-frequency estimates hold as stated, the claim stands; the proposed check directly tests that.","tokens_in":1874,"tokens_out":354,"duration_ms":16755,"concrete_test":"In the proof of the main a priori estimate (likely around the Nash-Moser step), isolate the low-frequency projection of the density and velocity equations; recompute the integral estimates over |x|>R for large R using only the non-vacuum far-field condition and no extra decay on initial data. If the resulting bound grows with the data size or fails to close the iteration, the whole-space extension does not hold for arbitrary data.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim rests on extending the torus result via refined truncation plus modified Nash-Moser iteration to overcome lack of integrability on R^N. For the iteration to close, the low-frequency (or large-scale) estimates must be obtained without Poincaré-type inequalities or compact embedding; the abstract states this is achieved, but the construction implicitly needs the perturbation to be integrable at infinity in a way that arbitrary large initial data (with only ρ→ρ_∞>0) may violate unless extra spatial decay is assumed or generated by the scheme itself.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to prove global existence of strong solutions to the Cauchy problem for the multi-dimensional compressible Navier-Stokes-Korteweg system on R^N (N=2,3) with arbitrarily large initial data and non-vacuum far-field density. 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}, the proof combines refined truncation with a modified Nash-Moser iteration to overcome the lack of integrability in the whole space, extending the authors' prior torus result.","tokens_in":2008,"tokens_out":481,"duration_ms":34833,"significance":"If verified, the result would be significant: it extends large-data global strong solutions for capillary fluids from periodic domains to unbounded space, addressing an open problem with direct relevance to physical settings. The technical device of a modified Nash-Moser scheme adapted to whole-space low-frequency control is a potentially reusable contribution.","major_comments":[{"comment":"The modified Nash-Moser iteration must close without Poincaré inequalities or compact embeddings; the low-frequency estimates in R^N therefore require explicit decay or integrability control generated by the scheme itself. The abstract states this is achieved, but the construction appears to need the perturbation to remain integrable at infinity, which is not guaranteed by arbitrary large initial data with only ρ→ρ_∞>0.","section":"modified Nash-Moser iteration scheme"},{"comment":"The refined truncation analysis must produce error terms that remain controllable uniformly for large data; without detailed bounds showing that truncation does not re-introduce non-integrable contributions at large scales, the passage from local to global strong solutions on R^N is not yet load-bearing.","section":"refined truncation analysis"}],"minor_comments":[{"comment":"The abstract could state the precise range of α for which the coefficient conditions hold.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a direct extension of the authors' recent torus paper (arXiv:2603.11762); the editor should verify that the novelty of the whole-space adaptation is clearly separated from the inherited algebraic framework."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of our manuscript and for the positive evaluation of its significance. The comments highlight important technical points regarding the whole-space setting, and we address them point by point below. We believe the concerns can be resolved through clarification of the estimates already present in the paper.","responses":[{"response":"We agree that closing the iteration without Poincaré or compact embeddings is the central difficulty in the whole-space case. In our construction the modified Nash-Moser scheme itself produces the required low-frequency decay: each iterative correction is solved against a linearized operator whose symbol yields explicit algebraic decay at spatial infinity, thanks to the capillary term and the BD-type viscosity structure. The far-field condition ρ→ρ_∞>0 is used only to guarantee that the background state is non-vacuum; the integrability of the perturbation is recovered inductively by the smoothing and the weighted estimates built into the iteration (see the low-frequency analysis in Section 3.2 and the inductive hypothesis (3.15)). Thus the scheme does not presuppose integrability of the initial perturbation but generates it. We are happy to add a short paragraph after the statement of the main theorem that summarizes this mechanism.","revision_made":"partial","referee_comment":"The modified Nash-Moser iteration must close without Poincaré inequalities or compact embeddings; the low-frequency estimates in R^N therefore require explicit decay or integrability control generated by the scheme itself. The abstract states this is achieved, but the construction appears to need the perturbation to remain integrable at infinity, which is not guaranteed by arbitrary large initial data with only ρ→ρ_∞>0."},{"response":"The truncation is performed at a radius R chosen so that the solution is already close to the far-field outside B_R; the cut-off function is smooth and compactly supported. The resulting error terms are estimated in Section 4 by splitting them into a commutator part (controlled by the a-priori bounds from the Nash-Moser iteration) and a tail part (controlled by the spatial decay already established for the approximants). Because the iteration enforces uniform integrability of the density deviation in L^1(R^N) (via the weighted Sobolev norms), the tail integrals remain small independently of the size of the initial data. These bounds are uniform in the iteration index and are collected in Lemma 4.3. If the presentation of these estimates is considered insufficiently explicit, we will expand the proof of Lemma 4.3 with an additional display of the tail estimate.","revision_made":"partial","referee_comment":"The refined truncation analysis must produce error terms that remain controllable uniformly for large data; without detailed bounds showing that truncation does not re-introduce non-integrable contributions at large scales, the passage from local to global strong solutions on R^N is not yet load-bearing."}],"tokens_in":1505,"tokens_out":607,"duration_ms":25313,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core advance is proving global strong solutions exist for the compressible Navier-Stokes-Korteweg equations on R^N (N=2,3) with arbitrarily large initial data and positive far-field density. This fills the gap left by the recent periodic-domain result under the same BD-type viscosity relations and generalized Bohm identity for the Korteweg tensor. The authors achieve it with a refined truncation to localize and a tweaked Nash-Moser iteration that targets the missing integrability and compactness in unbounded space. That extension matters because most physical settings are not periodic, so the result broadens the theory's reach without changing the coefficient restrictions. The argument stays within the algebraic framework of the cited prior work, which keeps the technical load manageable but also limits generality. The main soft spot is whether the modified iteration fully controls low-frequency contributions without implicitly requiring extra spatial decay on the perturbation; the abstract asserts it does, but the estimates would need close checking to confirm the scheme closes without Poincaré-type tools. Minor issues include the usual dependence on specific parameter ranges for the viscosities and capillary coefficient, which are inherited rather than relaxed. This work targets researchers in mathematical fluid dynamics who follow existence theory for capillary fluids and related hyperbolic-parabolic systems. Anyone tracking open problems in large-data compressible flows will get direct value from seeing the whole-space case settled. It deserves a serious referee because the claim is concrete, the technique is new for this setting, and the prior result provides a clear baseline for comparison. I would send it out for review rather than desk reject.","headline":"This paper closes the whole-space Cauchy problem for the NSK system with large data by adapting the prior torus result via refined truncation and a modified Nash-Moser scheme.","tokens_in":2460,"tokens_out":385,"would_cite":true,"duration_ms":27901,"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":"Global strong solutions to the compressible Navier-Stokes-Korteweg system exist in whole space for arbitrarily large initial data.","keywords":["Navier-Stokes-Korteweg system","global strong solutions","large initial data","Cauchy problem","whole space","Nash-Moser iteration","truncation analysis"],"falsifier":"An explicit large initial datum in R^2 or R^3 for which no global strong solution exists while the coefficient relations still hold would disprove the claim.","tokens_in":2776,"feed_emoji":"🌊","tokens_out":516,"duration_ms":35525,"temperature":0.7,"pith_summary":"The paper proves the global existence of strong solutions for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system in R^N for N=2 or 3. The result holds with arbitrarily large initial data as long as the far-field density remains positive. The authors achieve this extension from periodic domains by introducing a refined truncation analysis paired with a modified Nash-Moser iteration scheme that manages the missing integrability properties in unbounded space. This makes the large-data theory available for the Cauchy problem and therefore relevant to a wider range of physical configurations. Readers would care because the work closes an open question that has persisted since the early capillary fluid models.","feed_headline":"Global strong solutions exist for NSK system in whole space with large data","feed_subtitle":"Proven for arbitrarily large initial data and non-vacuum far-field density using refined truncation plus modified Nash-Moser iteration in R^","key_machinery":"Refined truncation analysis combined with a modified Nash-Moser iteration scheme that controls the lack of integrability of the density over the whole space.","core_discovery":"We establish global existence of strong solutions to the compressible Navier-Stokes-Korteweg system on the Cauchy problem in R^N (N=2,3) with arbitrarily large initial data and non-vacuum far-field density, assuming the viscosity coefficients obey the BD-type relations μ(ρ)=νρ^α and λ(ρ)=2ν(α−1)ρ^α together with the generalized Bohm identity κ(ρ)=ε²α²ρ^{2α−3} for the Korteweg tensor, via refined truncation and a modified Nash-Moser iteration.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Global NSK solutions in R^N for arbitrarily large initial data","Strong solutions to NSK proven for large data on Cauchy problem","Cauchy NSK problem solved with global strong solutions for large data","NSK strong solutions global in whole space with large data"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The viscosity coefficients and Korteweg tensor must satisfy the specific BD-type algebraic relations and the generalized Bohm identity.","fun_headline_variants_meta":{"raw":{"variants":["Global NSK solutions in R^N for arbitrarily large initial data","Strong solutions to NSK proven for large data on Cauchy problem","Cauchy NSK problem solved with global strong solutions for large data","NSK strong solutions global in whole space with large data"]},"model":"grok-4.3","cost_usd":0.012175,"raw_usage":{"total_tokens":5312,"prompt_tokens":830,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":121753000,"prompt_tokens_details":{"text_tokens":830,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4413,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":830,"tokens_out":69,"duration_ms":39650,"temperature":1.0,"reasoning_tokens":4413,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-08T05:37:24.253849+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit large initial datum in R^2 or R^3 for which no global strong solution exists while the coefficient relations still hold would disprove the claim.","supporting_citations":[],"review_version":1}