{"id":"2deb8d67-4a14-4257-b30d-ace735b546e0","arxiv_id":"2607.03602","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Very weak suitable Navier-Stokes solutions become Leray solutions once the velocity lies in a local Morrey space M^{p,γ} satisfying γ/p − 3/p + 2/3 < 0.","lead":"The paper defines very weak suitable solutions of the 3D Navier-Stokes equations and proves that membership of the velocity in certain local Morrey spaces forces them to be Leray solutions. This unifies and extends earlier criteria that used Lebesgue, Lorentz or classical Morrey spaces.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript supplies complete elementary proofs of the local-energy estimates, the pressure characterization, and the embeddings into local Morrey spaces. The scaling restriction that appears is forced by the method and is correctly stated. The continuity assumption singled out by the reader is indeed load-bearing for the limit R\to∞, yet it is openly hypothesized rather than tacitly assumed; once granted, the argument is free of circularity. Consequently the ACCEPT verdict with low correctness risk stands. The suggested test merely checks whether the hypothesis can be relaxed, not whether the existing proof is flawed.","tokens_in":20379,"tokens_out":418,"duration_ms":3957,"concrete_test":"Re-derive the passage from the localized inequality (21) to the global energy inequality (3) while replacing the strong continuity at t=0 by mere weak continuity in L^{2}; if the liminf argument still recovers (3) for every R\to∞ under the same Morrey condition (8), the continuity hypothesis can be weakened without affecting the claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption call (a-priori weak L^{2}_loc continuity for t>0 and strong continuity at t=0) is correctly identified as the technical hinge that lets the cut-off local-energy inequality pass to the global energy inequality (3). Within the paper's own definitions, however, that hypothesis is not an unstated gap: it is listed explicitly as Theorem 1.1(1) and is used only after the suitability measure μ and the Morrey control (8) have already been assumed. The subsequent estimates (Lemmas 4.1–4.2) close cleanly under the scaling condition γ/p-3/p+2/3<0, and the pressure identification (Proposition 1.1) is independent of suitability. No hidden circularity or scaling inconsistency appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces very weak suitable solutions of the 3D Navier–Stokes equations: distributional solutions with u in L^{3}_loc and P in L^{3/2}_loc such that the local energy defect measure μ defined by (6) is a non-negative locally finite measure. Theorem 1.1 states that if, in addition, u_{0} ∈ L^{2}(ℝ^{3}), the map t ↦ u(t,·) is weakly continuous in L^{2}_loc for t > 0 and strongly continuous at t = 0, and u belongs to the local Morrey space M^{p,γ}_{t,x}([0,T]×ℝ^{3}) for 0 < γ < 3 ≤ p < ∞ satisfying γ/p − 3/p + 2/3 < 0, then (u,P) is a Leray solution on [0,T] (and hence globally by the classical Leray program). The proof proceeds by testing the local energy inequality with carefully chosen cut-offs (20), obtaining the local energy estimates of Lemmas 4.1–4.2, and letting the spatial radius R → ∞ under the scaling condition (8). Proposition 1.1 identifies the pressure via Riesz transforms under a weaker Morrey assumption, and three corollaries give uniqueness and smoothness under an extra Morrey condition (16). Embeddings into Lebesgue, Lorentz, homogeneous Morrey and parabolic Morrey spaces are recorded in the appendix.","tokens_in":20544,"tokens_out":879,"duration_ms":6796,"significance":"The result supplies a clean, scale-invariant sufficient condition that upgrades a very weak suitable solution to a Leray solution inside a functional class strictly larger than the classical L^p_t L^q_x spaces used in earlier works (Foias, Fabes–Jones–Riviere, Giga, Galdi, Ding–Tan). Local Morrey spaces simultaneously contain Lorentz, homogeneous Morrey and (for short times) parabolic Morrey spaces, so the theorem unifies several previously separate settings. The proofs are fully written out, rely only on standard Riesz-transform bounds and real interpolation of Morrey spaces, and contain no free parameters or circular normalizations. The corollaries on uniqueness and regularity, while conditional on an extra Morrey assumption, are natural and correctly deduced from classical Serrin-type criteria. The contribution is therefore a useful technical remark that enlarges the known range of spaces in which the very-weak-to-Leray passage holds.","major_comments":[],"minor_comments":[{"comment":"In the statement of Theorem 1.1 the range 3 ≤ p < 9/2 is only deduced later from the combination of 0 < γ < 3 and (8); it would help the reader if this restriction were stated explicitly already in the theorem.","section":null},{"comment":"Lemma 1.1 requires T ≤ 2 for a purely technical reason (to guarantee √(T/2) ≤ 1 ≤ R). A short remark that the restriction is harmless because the local-to-global extension argument of Theorem 1.1 only needs a positive-time interval would clarify the scope.","section":null},{"comment":"Several typographical slips appear: “Cafarelli-Konh-Niremberg” (p. 3), “Theoem 5.2” (p. 15), and occasional missing spaces after commas. A light copy-edit would remove them.","section":null},{"comment":"In the proof of Lemma 4.1 the intermediate exponent is written “2/q” instead of “2/p”; the subsequent algebra is correct, but the notation should be consistent.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a short, carefully written technical note that fits well in a specialized analysis journal. The novelty is modest (a natural extension of known local-energy arguments to local Morrey spaces) but the execution is solid and the result is useful. I see no reason to request major changes."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper does exactly what the title promises: it defines very weak suitable solutions (velocity in L3_loc, pressure in L3/2_loc so the local energy inequality makes distributional sense, plus non-negative measure µ) and proves that local Morrey membership M^{p,γ} with the natural scaling γ/p − 3/p + 2/3 < 0, plus the usual L2 initial data and a continuity hypothesis, turns them into Leray solutions on [0,T]. The cut-off argument (test function α_{t0,t1,ε} φ_R, then R\to∞) is standard but carefully executed; Lemmas 4.1–4.2 close cleanly by Hölder and the Morrey norm, and the pressure identification (Prop. 1.1) via Riesz transforms on weighted L^p is independent and useful on its own.\n\nWhat is actually new is the very-weak-suitable class itself and the systematic use of local Morrey spaces as the ambient setting. The embeddings recover the classical L^p_t L^q_x, Lorentz, homogeneous Morrey and (for T≤2) parabolic Morrey criteria that appear in Foias, Fabes–Jones–Rivière, Giga, Galdi, Ding–Tan, etc. The three corollaries on uniqueness/regularity under an extra Morrey assumption are straightforward but correctly derived from Galdi’s theorems once the energy inequality is in hand.\n\nThe only real soft spot is the a-priori continuity hypothesis in Theorem 1.1(1): weak L2_loc continuity for t>0 and strong continuity at t=0. It is stated explicitly and is needed to pass to the limit after testing, so it is not hidden; still, it is an extra assumption that pure very-weak solutions need not satisfy. The range ends up restricted to 3≤p<9/2, and the T≤2 restriction for parabolic Morrey is purely technical. None of this breaks the argument.\n\nThe math is elementary functional analysis done carefully; citations are appropriate and non-circular. This is for people who already work on weak-solution theory for Navier–Stokes and care about the precise functional setting that upgrades very-weak to Leray. It deserves a serious referee and should appear as a short note. I would bring it to reading group only if we are deep in Morrey-space fluid dynamics that week; otherwise it is a useful reference rather than a must-read.","headline":"Clean technical note that upgrades very-weak suitable solutions to Leray solutions under local Morrey control; solid unification of known criteria, not a breakthrough.","tokens_in":21183,"tokens_out":621,"would_cite":true,"duration_ms":12275,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35B30","35B45"],"pacs":[],"model":"grok-4.5","headline":"Very weak suitable Navier-Stokes solutions become Leray solutions under a local Morrey bound on velocity.","keywords":["Navier-Stokes equations","very weak solutions","suitable solutions","Leray solutions","local Morrey spaces","local energy inequality","uniqueness","regularity"],"falsifier":"Exhibit a very weak suitable solution that lies in the stated local Morrey space, has L² initial data, yet fails to satisfy the global energy inequality (or fails to be weakly continuous in L²_loc).","tokens_in":21256,"feed_emoji":"🌊","tokens_out":720,"duration_ms":4996,"temperature":0.7,"pith_summary":"The paper asks when a very weak solution of the three-dimensional Navier-Stokes equations is actually a Leray solution: an energy-class weak solution that satisfies the global energy inequality. It introduces “very weak suitable solutions,” which satisfy only the minimal integrability needed to write the local energy inequality in the distributional sense and for which the associated defect measure is non-negative. Within this class the author proves that membership of the velocity in a local Morrey space M^{p,γ} (with 3 ≤ p < 9/2 and a decay condition γ/p − 3/p + 2/3 < 0), together with L² initial data and a mild continuity-in-time assumption, is enough to recover the global energy inequality. Local Morrey spaces contain the classical Ladyzhenskaya–Prodi–Serrin spaces as well as Lorentz, homogeneous Morrey and parabolic Morrey spaces, so the result unifies and extends several earlier criteria. Once the solution is known to be Leray, additional local Morrey assumptions yield uniqueness and smoothness on a possibly smaller time interval.","feed_headline":"Local Morrey bound turns weak NS solutions into Leray ones","feed_subtitle":"Suitable very-weak solutions with a mild decay condition recover the global energy inequality","key_machinery":"The local energy inequality tested against carefully chosen cut-off functions that are radial in space and approximate the indicator of a time interval; after integration by parts the non-negative defect measure and the local Morrey control make every remainder vanish as the spatial cut-off radius tends to infinity, yielding the global energy inequality.","core_discovery":"A very weak suitable solution (u,P) of the Navier–Stokes equations on [0,T]×ℝ³ becomes a Leray solution whenever the initial velocity lies in L², the map t ↦ u(t,·) is weakly continuous in L²_loc for t>0 and strongly continuous at t=0, and u belongs to the local Morrey space M^{p,γ}_{t,x} for parameters satisfying 0<γ<3≤p<∞ and γ/p − 3/p + 2/3 <0.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Local Morrey bound upgrades very-weak NS to Leray solutions","Mild local Morrey condition turns suitable weak NS into Leray","Very-weak suitable NS solutions become Leray under Morrey bound","Local Morrey spaces recover Leray status for weak NS flows","Suitable very-weak NS solutions yield Leray via local Morrey"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The velocity must already be weakly continuous in L² on every compact set for positive times and strongly continuous at time zero; without this continuity the passage from the local energy inequality to the global energy inequality fails.","fun_headline_variants_meta":{"raw":{"variants":["Local Morrey bound upgrades very-weak NS to Leray solutions","Mild local Morrey condition turns suitable weak NS into Leray","Very-weak suitable NS solutions become Leray under Morrey bound","Local Morrey spaces recover Leray status for weak NS flows","Suitable very-weak NS solutions yield Leray via local Morrey"]},"model":"grok-4.5","effort":"low","cost_usd":0.004796,"raw_usage":{"total_tokens":1322,"prompt_tokens":736,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":47960000,"prompt_tokens_details":{"text_tokens":736,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":513,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":736,"tokens_out":73,"duration_ms":4866,"temperature":1.0,"reasoning_tokens":513,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T01:15:07.682822+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a very weak suitable solution that lies in the stated local Morrey space, has L² initial data, yet fails to satisfy the global energy inequality (or fails to be weakly continuous in L²_loc).","supporting_citations":[],"review_version":1}