{"id":"8fbc0926-4dc5-45e2-84b2-4de6213e620e","arxiv_id":"2607.04918","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Unique global Navier-Stokes solutions exist when horizontal velocity is small in critical Besov spaces and vertical velocity is arbitrarily large in the ill-posed space Ḃ^{-1}_{∞,σ}.","lead":"The paper proves unique global solutions exist for 3D Navier-Stokes when the horizontal velocity starts small in critical Besov spaces, even if the vertical velocity is large in an endpoint space known to make the equations ill-posed. This enlarges the set of initial data for which global regularity is guaranteed without Gevrey-class restrictions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim of Theorem 1.1 is a carefully delimited extension of the critical-space theory: horizontal data small in Ḃ^{3/p-1}_{p,q} (3/2<p<3) while the vertical component may be large in the ill-posed space Ḃ^{-1}_{\\infty,\theta}. The proof follows a standard Banach-fixed-point + a-priori-estimate route once the system is rewritten via div-free condition. All nonlinear estimates are reduced to para-product bounds that are classical once the indices stay inside the open range stated in Lemmas 2.2–2.4; the only genuinely restrictive condition is p<3, which the author openly acknowledges. The time-decomposition device (Lemma 2.1) is correctly applied and produces the exponential smallness factor. Because the manuscript supplies every estimate needed to check the argument and no internal inconsistency surfaces, the reader's ACCEPT verdict with high confidence stands. The single verification step proposed above simply confirms that the most delicate product estimate is not mis-indexed; if it holds, the theorem is solid.","tokens_in":18356,"tokens_out":546,"duration_ms":4568,"concrete_test":"Independently re-derive the product estimate for the non-divergence term that appears in the third display of Lemma 2.2 (and the corresponding term in Lemma 2.4) for a single fixed triple (p,r,\rho) inside the open range 3/2<p<3, 2<r<\rho^{-1}<2p/3; if the Besov indices fail to satisfy the conditions of Corollary A.3, the claimed bound does not hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (p<3 forced by the non-divergence term u3\nabla h·uh) is correctly identified and is already stated as essential by the author (discussion after (1.6) and ranges of Lemmas 2.3–2.4). Within the claimed parameter regime the estimates close: the time-decomposition Lemma 2.1 reduces the large vertical norm to small pieces, the para-product bounds of Lemmas 2.2–2.4 control every nonlinear term that appears after the divergence-free rewriting (1.4)–(1.5), and the a-priori estimate of Lemma 3.2 together with the fixed-point construction of Lemma 3.1 yield global existence under the exponential smallness (1.3). No hidden gap, circularity, or range violation appears in the argument as written.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves global well-posedness for the 3D incompressible Navier–Stokes equations when the horizontal velocity components a_h are small in the critical Besov space ḎB^{3/p-1}_{p,q} (3/2 < p < 3, 1 ≤ q ≤ 2σ) while the vertical component a_3 may be arbitrarily large in the endpoint space ḎB^{-1}_{∞,σ} (1 ≤ σ < ∞). After rewriting the system via the divergence-free condition so that the equation for u_3 becomes linear in the vertical velocity, the author establishes bilinear estimates in Chemin–Lerner spaces (Lemmas 2.2–2.4), obtains a local solution by contraction (Lemma 3.1), derives an a-priori bound that interpolates the vertical norms (Lemma 3.2), and extends the solution globally by a time-decomposition argument (Lemma 2.1) under the exponential smallness condition (1.3). The resulting solution belongs to the natural energy spaces E_{p,q}(0,∞)^2 \times E_{∞,σ}(0,∞) with the stated bounds.","tokens_in":18522,"tokens_out":772,"duration_ms":5451,"significance":"The result is a genuine advance: it produces unique global solutions whose vertical component lies in a space where the full Navier–Stokes system is known to be ill-posed, without requiring Gevrey regularity or smallness of a_3 in a stronger critical space. The comparison with Chemin–Gallagher–Paicu and with Iwabuchi–Nakamura is accurate and the example of initial data (Remark 1.2(4)) shows that the theorem covers data outside the reach of previous theories. The technical ingredients—para-product estimates adapted to mixed horizontal/vertical norms and the time-decomposition lemma—are cleanly executed and of independent interest for anisotropic or partially large-data problems.","major_comments":[],"minor_comments":[{"comment":"Acknowledgements: the Grant Number is written as the list of keywords rather than an actual KAKENHI number; this should be corrected.","section":null},{"comment":"Page 1, line after (1.2): “Leter” should be “Later”; several other minor typos appear (e.g., “estiamtes”, “nonlinearterms”).","section":null},{"comment":"Lemma 3.1 is stated without proof; a one-sentence reference to the standard fixed-point argument via Lemmas 2.2–2.4 would improve readability.","section":null},{"comment":"In the definition of N in the proof of Theorem 1.1 the floor function is applied to a quantity that already contains the large vertical norm; a brief remark that N remains finite under the exponential smallness (1.3) would make the contradiction argument more transparent.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, self-contained and technically solid. It is a natural fit for a strong analysis journal. No concerns about novelty disclosure or citation pattern."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper gives a clean global existence result for 3D Navier-Stokes that is new in the critical-space literature: horizontal velocity small in Ḃ^{3/p-1}_{p,q} (3/2 < p < 3), vertical component allowed large in Ḃ^{-1}_{∞,σ} (σ < ∞) under only an exponential smallness factor. That is exactly the space where the full system is known to be ill-posed, so the mixed-norm statement is the real advance.\n\nWhat works well is the structure. After the divergence-free rewrite (1.4)–(1.5) the vertical equation becomes linear in u3, so the dangerous quadratic terms disappear. The author then controls the remaining products with standard para-product estimates in Chemin-Lerner spaces (Lemmas 2.2–2.4) and uses the time-decomposition lemma (from his earlier work) to chop the large vertical norm into small pieces on successive intervals. Local existence is ordinary contraction; the a-priori bound closes under the exponential condition and yields global solutions. The argument is carefully written, the ranges are stated honestly, and the self-citations are only for a technical lemma whose statement is independent of the present claim.\n\nThe soft spots are real but already flagged by the author. The restriction p < 3 is forced by the non-divergence term u3 ∇h · uh; for p ≥ 3 the product estimates fail and the argument stops. The endpoint σ = ∞ is also excluded. Both limitations are proportional: they shrink the result but do not break it inside the claimed regime. No circularity, no hidden parameters, no range violations appear on a full reading.\n\nThis is for people who work on critical Besov theory for NS or anisotropic systems. Specialists will check the bilinear estimates in a few hours; the rest of the community can take the theorem as a legitimate extension of the Fujita–Kato / Koch–Tataru program. I would send it to a serious referee without hesitation.","headline":"Solid mixed-norm global well-posedness for NS: horizontal small in critical Besov, vertical large in the ill-posed endpoint class, via anisotropic rewrite + time decomposition.","tokens_in":19161,"tokens_out":506,"would_cite":true,"duration_ms":4607,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76D05"],"pacs":[],"model":"grok-4.5","headline":"3D Navier-Stokes admits unique global solutions when only the horizontal velocity is small in critical Besov spaces, even if the vertical velocity is large in the ill-posed endpoint space.","keywords":["Navier-Stokes equations","critical Besov spaces","global well-posedness","large vertical velocity","Chemin-Lerner spaces","time decomposition","ill-posed endpoint"],"falsifier":"Exhibit a divergence-free initial datum with horizontal part small in the stated Besov space and vertical part large in ḊB^{-1}_{∞,σ} for which the corresponding mild solution either blows up in finite time or fails to remain unique in the Chemin-Lerner class.","tokens_in":19235,"feed_emoji":"🌊","tokens_out":725,"duration_ms":5107,"temperature":0.7,"pith_summary":"The three-dimensional incompressible Navier-Stokes equations are known to be well-posed for small data in critical spaces, yet ill-posed when the data are merely large in the endpoint Besov space of order minus one. This paper shows that the ill-posedness can be avoided by treating the components differently: if the two horizontal components of the initial velocity are sufficiently small in a classical critical Besov space, the vertical component may be arbitrarily large in that same endpoint space and a unique global solution still exists. The argument rewrites the system so that the vertical equation becomes linear in the vertical velocity, then controls the remaining nonlinear interactions by a time-interval decomposition that makes the large vertical field small on successive pieces. The result therefore enlarges the set of initial data for which global regularity is guaranteed, without Gevrey-class assumptions or smallness of the vertical part.","feed_headline":"Large vertical velocity still yields global Navier-Stokes flow","feed_subtitle":"Only the horizontal components need to be small in critical spaces; the vertical part may sit in an ill-posed class","key_machinery":"Divergence-free rewriting of the system into a horizontal equation and a vertical equation that is linear in the vertical velocity, closed by para-product estimates and a time-decomposition lemma that makes the large vertical field small on successive time intervals.","core_discovery":"Under the parameter range 3/2 < p < 3, 1 ≤ σ < ∞ and suitable q, r, θ, any divergence-free initial velocity whose horizontal part satisfies a smallness condition of the form ∥a_h∥ exp(C ∥a_3∥^{r/θ}) ≤ η admits a unique global solution in the corresponding Chemin-Lerner spaces, with the vertical velocity allowed to be large in ḊB^{-1}_{∞,σ}.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Global NS solutions allow large vertical velocity in ill-posed Besov class","Small horizontal velocity yields unique global NS flow with large vertical part","Global Navier-Stokes exists despite large vertical data in Ḃ_∞,σ^{-1}","Horizontal smallness alone guarantees global unique NS solutions","Large vertical NS component in ill-posed space still gives global flow"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The horizontal integrability exponent must stay strictly less than three; otherwise the product estimate for the non-divergence-form term that multiplies vertical velocity by the horizontal divergence fails and the a-priori bounds no longer close.","fun_headline_variants_meta":{"raw":{"variants":["Global NS solutions allow large vertical velocity in ill-posed Besov class","Small horizontal velocity yields unique global NS flow with large vertical part","Global Navier-Stokes exists despite large vertical data in Ḃ_∞,σ^{-1}","Horizontal smallness alone guarantees global unique NS solutions","Large vertical NS component in ill-posed space still gives global flow"]},"model":"grok-4.5","effort":"low","cost_usd":0.0065,"raw_usage":{"total_tokens":1572,"prompt_tokens":685,"num_sources_used":0,"completion_tokens":95,"cost_in_usd_ticks":65000000,"prompt_tokens_details":{"text_tokens":685,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":792,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":685,"tokens_out":95,"duration_ms":6154,"temperature":1.0,"reasoning_tokens":792,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T11:27:13.742561+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a divergence-free initial datum with horizontal part small in the stated Besov space and vertical part large in ḊB^{-1}_{∞,σ} for which the corresponding mild solution either blows up in finite time or fails to remain unique in the Chemin-Lerner class.","supporting_citations":[],"review_version":1}