{"id":"da56f610-37be-4d47-b23a-5028f29611d9","arxiv_id":"2506.22235","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Corrected ν-independent weighted estimates for the 2D Navier-Stokes system with large bulk viscosity give global regularity with vacuum and an explicit ν^{-1/2} incompressible-limit rate.","lead":"These authors fix a flawed estimate in prior work on the 2D compressible Navier-Stokes equations and prove that, when bulk viscosity is large, global smooth solutions can exist even if the fluid has vacuum regions. They also show the fluid approaches an incompressible flow with an explicit rate as the bulk viscosity grows.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The entire new estimate (1.11) relies on Lemma 3.1, an unverified black-box import of [10, Props 3.1–3.2]; if those propositions secretly depend on the flawed Prop 3.3, the corrected proof and the ν^{-1/2} rate collapse.","rationale":"The reader's weakest-assumption field identifies exactly the same load-bearing concern: Lemma 3.1 takes [10, Propositions 3.1 and 3.2] as black boxes, and every new estimate is built on them. My independent reading confirms this is the most critical point. I also scanned the internal estimates: the proof of Lemma 3.2 appears structurally sound apart from minor typographical issues (e.g., (3.32) writes ∥√ρ˙u∥^4_{L4} where the surrounding estimates suggest ∥√ρ˙u∥^4_{L2}; (3.17) omits a 1/μ factor; the β constant in (4.38) is off by a small factor). These are fixable and do not undermine the argument. The deferred proof of Theorem 1.1 is a real presentational gap, but it is secondary to the black-box dependency. Therefore I do not change the reader's conditional verdict, but I recommend that acceptance be granted only after the concrete check on [10, Props 3.1–3.2] confirms their independence from Proposition 3.3.","tokens_in":26102,"tokens_out":37706,"duration_ms":335386,"concrete_test":"Read the published [10] and trace the proofs of Propositions 3.1, 3.2, and inequality (3.47) line by line. Check whether any of their intermediate steps invokes Proposition 3.3, the t-weighted estimate (3.64), or the erroneous identity (1.7). If any such dependency appears, Lemma 3.1 is invalid and the present paper's central claims fail; if no dependency appears, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—the corrected ν-independent t-weighted estimate (1.11), hence Proposition 1.2, Theorem 1.1, and the convergence rate (1.15)–(1.16)—is built on Lemma 3.1, which is stated as a summary of [10, Propositions 3.1, 3.2] and [10, (3.47)] but is not re-proved. The current paper's purpose is precisely to repair the flaw in [10, Proposition 3.3], so the independence of the imported propositions from Proposition 3.3 is load-bearing. If [10, Props 3.1–3.2] were themselves derived using the erroneous identity (1.7), the claimed ν-independent global bounds (3.1)–(3.2), or any t-weighted estimate of type (1.5), then Lemma 3.1 is invalid and the present proof has no foundation. Remark 1.1 asserts that the fixed-ν existence results of [10] remain valid, which is plausible but is not a proof of the ν-independence of the imported propositions. Moreover, Theorem 1.1 is not proved here; its proof is deferred to [10, Sections 4 and 5] with no adaptation shown, and those sections were built around the flawed t-weighted estimate. This is the single most load-bearing soft spot: without Lemma 3.1, the new inequalities and the convergence rate lose their basis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the 2D barotropic compressible Navier–Stokes equations on the torus with large bulk viscosity ν = 2µ + λ and possibly vanishing density. It identifies an error in Danchin–Mucha [10, Proposition 3.3]: the identity (1.7) used there replaces ∇(tr(∇u·∇u)) by ∇(tr(∇u·∇Qu)), and the paper shows that the correct commutator identity (1.8) contains the genuinely different term ∇(tr(∇u·∇u)). The paper then proves a modified ν-independent weighted estimate, Proposition 1.2 / (1.11), in which the material derivative of the divergence, V̇ = D_t(divu), and the curl ∇⊥·˙u replace the quantities appearing in (1.5); the proof isolates the difficult quadratic term N₆ in (3.13) and controls it through the effective viscous flux F and Hardy-space duality. From (1.11) it derives (1.12): (∇⊥·u, divu) ∈ L^{2−ε}(0,T;L∞) for any ε ∈ (0,1). Theorem 1.1 states global existence and uniqueness, with the proof deferred to [10, Sections 4 and 5]. Theorem 1.2 gives the incompressible-limit rate (1.15)–(1.16): for vacuum-allowed solutions, sup(∥√η(Pu−v)∥² + ∥ρ−η∥²) + ∫(∥Pu−v∥² + ∥∇(Pu−v)∥²)dt ≤ Cν^{−1/2}, based on t-growth and singular t-weighted estimates for φ, ϕ and a weighted L² estimate for Pu−v.","tokens_in":26397,"tokens_out":45043,"duration_ms":401741,"significance":"If the main estimates hold, the paper provides a genuine repair of a published proof, supplies the ν-independent global bounds that the Danchin–Mucha framework needs, and proves the first convergence rate for the incompressible limit in the presence of vacuum, extending [7,9] where vacuum was excluded. The strength of the manuscript is its detailed, explicit energy analysis: Lemmas 3.2 and 3.3 are proved with displayed absorption constants, the cancellation in (3.33)–(3.35) using the effective viscous flux and the Coifman–Lions–Meyer–Semmes Hardy-space estimate is convincing, and the convergence-rate proof in Section 4 is internally consistent, with a correct Gronwall strategy and correctly tracked powers of ν. I verified the main differential identities (3.13), (4.16)–(4.18) and the absorption/Young steps, including the delicate factor ν^{−2/3} in (4.37)–(4.38). The reservations expressed below concern external support and deferred proofs rather than the internal algebra of the new estimates.","major_comments":[{"comment":"The paper's foundational input is a black box. Every new estimate in the paper relies on Lemma 3.1: Lemma 3.2 invokes (3.1)–(3.2) in nearly every displayed bound of Steps 2–4, Lemma 3.3 uses them in (3.39)–(3.40), and Theorem 1.2 rests on Lemmas 3.1–3.3. The proof of Lemma 3.1, however, is only the sentence 'combining [10, Propositions 3.1 and 3.2] with [10, (3.47)] proves (3.1) and (3.2),' and the content of [10, (3.47)] is not stated. Since the paper's purpose is to repair a flaw inside Section 3 of [10], the reader cannot verify from the manuscript that the imported ν-independent bounds of (3.2) — especially ∫(∥∇F∥²_{L²}+∥∇u∥⁴_{L⁴})dt ≤ C — do not use the erroneous identity (1.7). The paper's own narrative ('the largeness of ν was determined in [10, Propositions 3.1 and 3.2] and was used to derive' the t-weighted estimate) suggests those propositions precede [10, Proposition 3.3] and are probably unaffected, so the circularity concern from the stress-test note does not appear to be realized; nevertheless, the independence must be demonstrated in the manuscript, not inferred. The revision should state the imported propositions and [10, (3.47)] explicitly (or reproduce the proofs in an appendix) and indicate where each ingredient of (3.1)–(3.2) is proved, ruling out any use of the flawed identity.","section":"Section 3, Lemma 3.1"},{"comment":"The existence half of the first main theorem is deferred. The proof reads 'we can prove the global existence theorem and the uniqueness result stated in Theorem 1.1 in the same manner as that in [10, Sections 4 and 5],' with no adaptation shown. Because [10, Sections 4 and 5] were built around the t-weighted estimate (1.5) whose proof this paper declares flawed, the authors should verify that the construction, compactness, and uniqueness arguments in those sections go through with the corrected estimates (1.11)–(1.12) in place of (1.5); in particular, (1.12) provides (divu, ∇⊥·u) ∈ L^{2−ε}(0,T;L∞), which yields ∇u ∈ L^{2−ε}(0,T;BMO) by the div-curl structure, and the paper should confirm that this is the property actually consumed by the uniqueness argument in [10]. Remark 1.1 likewise asserts, without proof, that the proof of [10, Proposition 3.3] yields a ν-dependent version of (1.5) so that the fixed-ν existence theorem remains valid; this claim needs a proof or a precise citation once the published proof of that proposition has been identified as flawed.","section":"Section 3, proof of Theorem 1.1 and Remark 1.1"}],"minor_comments":[{"comment":"The displayed identity should read ∇⊥·(ρ˙u) = µ∆(∇⊥·u); a factor µ is missing, and the same factor is missing in the derived bounds (3.18) and (3.38). Since µ > 0 is fixed and the paper's constants are allowed to depend on µ, this does not affect the ν-uniformity of the claims, but the identities should be corrected.","section":"Equation (3.17)"},{"comment":"The last term in (3.32) is written as ∥√ρ˙u∥⁴_{L⁴}; from (3.19)–(3.31) it should be ∥√ρ˙u∥⁴_{L²}, and the displayed norm as written is not what the preceding estimates produce.","section":"Equation (3.32)"},{"comment":"Lemma 3.3 is stated for 2 < q < ∞, while Proposition 1.2(1.12) claims 2 ≤ q < ∞; the endpoint q = 2 follows from (3.1)–(3.3) by Hölder interpolation, and a one-line justification would reconcile the two statements.","section":"Lemma 3.3 vs Proposition 1.2"},{"comment":"There are several typos that should be corrected: 'baratropic compressible Naveir-Stokes' in Section 1, 'Propsition' in the heading of Proposition 1.1, 'Riersz transform' in Section 2, and 'valur problem' in reference [28].","section":"Throughout"},{"comment":"The tensor ∇⊥⊗u and the identity for I₁ are stated without derivation; a displayed verification or a reference would improve readability, since (3.6) is used to obtain the sign of the main energy dissipation term.","section":"Equations (3.6)–(3.7)"}],"recommendation":"major_revision","confidential_remarks":"The main decision-relevant risk is external dependency: Lemma 3.1 is imported from [10], and Theorem 1.1 is deferred to [10]. I would advise requiring the revision to make the imports auditable (e.g., in an appendix) or to restate the theorems with the imported bounds as explicit hypotheses. I also recommend that the authors be transparent, in the abstract and introduction, that the 'fix' of [10, Proposition 3.3] consists of a modified t-weighted estimate (1.11) rather than a proof of (1.5) as published, since the two left-hand sides differ in the terms ν∥div ˙u∥² versus ν∥V̇∥². The paper is otherwise within scope and the new estimates appear sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this paper fixes a real, concrete error in Danchin-Mucha's 2023 Compressible Navier-Stokes proof: the identity (1.7) misidentifies tr(\\nabla u \\cdot \\nabla Qu) with tr(\\nabla u \\cdot \\nabla u), and their t-weighted estimate (1.5) does not follow. The authors show the discrepancy, and they replace the failed estimate with a genuinely different one: (1.11), where the material derivative of div u replaces div of the material derivative. That is a substantive change, and the proof in Lemma 3.2 is detailed and mostly convincing. The high point is the treatment of N6, where the effective viscous flux and the Hardy-BMO duality are used to avoid the \\nu^{1/4}\\|\\nabla Pu\\| bound that the old argument needed. I believe the correction is real.\n\nSecond, the paper's new convergence-rate theorem (1.15)-(1.16) for the incompressible limit in the presence of vacuum is a true advance: to my knowledge no rate was known there, and the \\nu^{-1/2} rate is plausible given the machinery. The decomposition \\rho-\\eta = \\phi+\\tilde{\\phi}, the t-growth and singular t-weighted estimates, and the splitting of M8 are all reasonable.\n\nThe largest soft spot is structural. Lemma 3.1 is taken verbatim from [10, Props 3.1-3.2] and is not reproved. Since the entire corrected estimate builds on it, the independence of those propositions from the flawed Prop 3.3 is load-bearing. The authors seem to believe it (Remark 1.1), and it may well be true, but they do not show it, and a referee will need that checked. The same goes for Theorem 1.1, which is deferred to [10, Sections 4-5], and for the compactness part of the incompressible limit, which is stated as 'standard arguments' and omitted.\n\nThere are also smaller issues. Equation (3.17) drops the shear viscosity \\mu when writing \\Delta(\\nabla^\\perp \\cdot u) = \\nabla^\\perp \\cdot (\\rho \\dot u); since \\mu is a fixed constant this does not affect the \\nu-independence, but it is wrong as displayed. Some estimates cite (3.2)-(3.3) before Proposition 1.2 is stated, which is harmless but confusing. All minor.\n\nNet: this deserves a serious referee. My own verdict would be 'revise and re-read carefully,' with the main condition being a verification that [10, Props 3.1-3.2] do not depend on the flawed estimate. The math is honest, the error is real, and the new estimate is the right kind of repair. Send it out.","headline":"A genuine correction of a published t-weighted estimate, with a new vacuum convergence rate; refereeable, but the black-box import from the corrected paper needs verification.","tokens_in":26932,"tokens_out":4441,"would_cite":true,"duration_ms":44350,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76N10","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves global regularity and an explicit incompressible-limit rate for 2D compressible Navier-Stokes equations with large bulk viscosity, even when the density contains vacuum.","keywords":["compressible Navier-Stokes equations","global large solutions","vacuum","large bulk viscosity","incompressible limit","convergence rate","t-weighted estimates"],"falsifier":"A concrete verification: take a smooth solution family indexed by $\\nu$ with fixed initial data satisfying (1.9) and $\\operatorname{div}u_0$ small, and numerically monitor $\\int_0^T t(\\mu\\|\\nabla^\\perp\\cdot\\dot u\\|_{L^2}^2+\\nu\\|\\dot V\\|_{L^2}^2)\\,dt$. The central claim is that this quantity stays bounded as $\\nu\\to\\infty$; a sequence for which it grows would falsify Proposition 1.2 and with it the rate theorem.","tokens_in":25896,"feed_emoji":"🌊","tokens_out":9505,"duration_ms":92470,"temperature":0.7,"pith_summary":"The paper studies 2D barotropic compressible Navier-Stokes equations on the torus with large bulk viscosity $\\nu$, allowing the fluid density to vanish. Its central claim is that a certain $\\nu$-independent weighted estimate holds, provided one tracks the material derivative of the divergence, $\\dot V=\\partial_t(\\operatorname{div}u)+u\\cdot\\nabla\\operatorname{div}u$, rather than the divergence of the material derivative. This repairs an error in the corresponding estimate of [10, Proposition 3.3] and yields the regularity information $(\\nabla Pu,\\operatorname{div}u)\\in L^r(0,T;L^\\infty)$. As a by-product, the paper proves a convergence rate of order $\\nu^{-1/2}$ for the incompressible limit as $\\nu\\to\\infty$, a case that earlier rate results excluded because they required the density to stay away from vacuum.","feed_headline":"Incompressible limit of 2D viscous gas gains a proven rate with vacuum","feed_subtitle":"A corrected t-weighted estimate yields global regularity and an explicit convergence rate that earlier results could not reach in vacuum.","key_machinery":"The load-bearing object is the effective viscous flux $F=\\nu\\operatorname{div}u-(P(\\rho)-\\overline P)$, used in place of $\\nu\\operatorname{div}u$ because $\\nabla F$ is controlled in $L^2$ independently of $\\nu$, while $\\nu^{1/4}\\|\\nabla u\\|_{L^4}$ is not. The proof also pivots on estimating $\\dot V=\\partial_t(\\operatorname{div}u)+u\\cdot\\nabla\\operatorname{div}u$ rather than $\\operatorname{div}\\dot u$, which converts the problematic term $\\nu\\langle t\\dot u,\\nabla(\\operatorname{tr}(\\nabla u\\cdot\\nabla u))\\rangle$ into pieces that can be absorbed by the flux $F$. A third mechanism is the identity $\\partial_i u\\cdot\\nabla u_i=(\\operatorname{div}u)^2+2\\nabla^\\perp u_2\\cdot\\nabla u_1$: the first term is handled through $F$, and the Hardy-space/BMO duality estimate $\\|F\\nabla^\\perp u_2\\cdot\\nabla u_1\\|_{H^1}\\le \\|\\nabla F\\|_{L^2}\\|\\nabla u\\|_{L^2}^2$ controls the second.","core_discovery":"The central discovery is that the right quantity to control in the two-dimensional vacuum problem is $\\dot V=\\partial_t(\\operatorname{div}u)+u\\cdot\\nabla\\operatorname{div}u$. With the effective viscous flux $F:=\\nu\\operatorname{div}u-(P(\\rho)-\\overline P)$ and the solenoidal projector $P$, the paper proves that for $\\nu$ sufficiently large and any $T>0$, $\\sup_{0\\le t\\le T} t\\int\\rho|\\dot u|^2\\,dx+\\int_0^T t\\int(\\mu|\\nabla^\\perp\\cdot\\dot u|^2+\\nu|\\dot V|^2)\\,dx\\,dt\\le C(T)$, with $C(T)$ independent of $\\nu$. From this it derives $(\\nabla Pu,\\operatorname{div}u)\\in L^r(0,T;L^\\infty)$ for some $1