{"id":"1fb93fed-002e-4be6-bf79-2e57ddaa9e1e","arxiv_id":"2608.00727","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the 3D Navier-Stokes-Korteweg system with a small stationary force, the authors prove global uniform strong solutions and the low Mach convergence rate ε^{min(1/r, 1/2-1/p)} in mixed Besov spaces.","lead":"For the three-dimensional compressible Navier-Stokes-Korteweg equations with a small stationary external force, this paper proves global strong solutions for ill-prepared perturbations and a quantitative low Mach number convergence rate toward the incompressible Navier-Stokes flow. A specialist would read it for the Korteweg-symmetric spectral analysis that treats capillary acoustic waves, which are wave-like at low frequencies and Schrödinger-like at high frequencies.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The uniform low-frequency Besov estimate (4.19) is asserted without proof and does not follow from the preceding dyadic inequality; global existence and the convergence rate both depend on it.","rationale":"The paper aims to prove a low Mach number limit with an explicit rate for the NSK system around a nontrivial stationary flow. The most load-bearing step is the a priori estimate (4.19), because it is used to close the bootstrap for global existence and to derive the time decay that feeds into the acoustic source estimates and the final convergence rate. The reader's stated weakest assumption concerns the spectral projectors and the positivity condition l_ε > 0, which is a parameter restriction rather than a gap in the argument; I agree that assumption is structurally important, but it is not the most fragile part of the proof. In the reader's rationale, point (1) flags (4.19) as asserted rather than derived, and my analysis confirms that this is a genuine missing proof. The dyadic inequality (4.15) is not enough as written: after time integration, the low-frequency forcing terms do not obviously produce a uniform bound in ˙B^{1/2}_{2,∞}; they may contain a divergent 2^{-j} factor, and the source terms are only shown to be L∞_t-bounded, not L^1_t with suitable decay. This affects the central claim because without (4.19) the global existence and the rate (1.12) are not established. I also noted an index inconsistency in (4.12) and an ambiguity in the initial data of the limiting incompressible flow, but these are more easily fixable. Since the identified gap is a missing proof step rather than a demonstrated counterexample, the appropriate verdict remains CONDITIONAL: the paper should be accepted only if (4.19) is rigorously derived or an alternative low-frequency control is supplied.","tokens_in":36849,"tokens_out":22292,"duration_ms":184446,"concrete_test":"Write out the proof of (4.19) from (4.15) by integrating the dyadic inequality for j ≤ 0. Specifically, define a_j(t)=2^{j/2}E_j(t)^{1/2} and derive the bound a_j(t) ≤ a_j(0) + C sup_τ ||U(τ)||_{B^{3/2}_{2,∞}∩H^N} ∫_0^∞ (2^{j/2}||K^{1/2}H(τ)||_{B^{1/2}_{2,∞}} + ||G(τ)||_{B^{-1/2}_{2,∞}}) dτ, then check whether the right-hand side is bounded uniformly in j ≤ 0. If it contains a factor 2^{-j} or requires L^1_t bounds on H and G that are not established, then (4.19) does not follow from the stated lemmas and the proof of Theorem 1.2 is incomplete unless (4.19) is reproved by a different low-frequency mechanism.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.2 critically uses the uniform bound (4.19), sup_t ||(K^{1/2}σ_ε,w_ε)||_{B^{1/2}_{2,∞}} ≤ C ||initial||_{B^{1/2}}, to close the bootstrap, obtain time decay, and ultimately produce the rate in (1.12). However, (4.19) is stated as an immediate consequence of Lemmas 4.1–4.3, with no derivation. Lemma 4.3 gives the dyadic inequality (4.15): d_t E_j + c 2^{2j} E_j ≤ C 2^j ||K^{1/2}H_j||_{L^2} E_j^{1/2} + C ||G_j||_{L^2} E_j^{1/2}. To pass to the B^{1/2}_{2,∞} norm one multiplies by 2^{j/2} and integrates in time. Using only the L∞_t bounds from Lemmas 4.1–4.2, the integrated source contains a factor 2^{-j} for j ≤ 0 after the damping e^{-c2^{2j}t} is applied; this factor tends to infinity as j → -∞, so the dyadic sum does not converge uniformly unless additional low-frequency time integrability is proved. The paper does not provide such an argument, and the low-frequency part of H and G is not shown to lie in L^1_t with the needed weights. Since (4.19) is used before the decay estimates (4.21)–(4.23) are available, the circularity is not resolved by later decay. The same gap is implicitly acknowledged by the reader, who lists (4.19) as needing attention.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the low Mach number limit for the three-dimensional compressible Navier–Stokes–Korteweg system on R^3 with a small stationary external force. It first constructs stationary NSK solutions uniformly in the Mach number ε, proving that the stationary density fluctuation and the compressible velocity component are of order ε². For ill-prepared perturbations around these stationary profiles, the authors claim global strong well-posedness uniformly in ε and a quantitative convergence rate in mixed Besov norms: the acoustic part, the compressible velocity, and the incompressible projection error are of order ε^{min{1/r, 1/2 - 1/p}} in L^r_t B^s_{p,1}. The proof combines an elliptic fixed-point argument for the stationary problem, high-order energy estimates with a Kawashima-type cross term, a low-frequency Besov estimate, dyadic dispersive estimates for the acoustic–capillary semigroup, and a decomposition of the Duhamel source into time-integrable and stationary-coefficient parts.","tokens_in":37098,"tokens_out":24617,"duration_ms":211494,"significance":"If the main theorem is correct, the paper gives the first global-in-time low Mach number limit for a capillary compressible system around a nontrivial stationary flow with a general, non-potential stationary force. The stationary part improves on the non-capillary result of Deguchi by converting the density equation into an elliptic problem for the operator B = p'(ρ∞) - κρ∞Δ, yielding ε² accuracy for both density and velocity errors. The nonstationary part contains a genuinely novel structural point: in Korteweg-symmetric variables the spectral projectors are uniformly bounded zero-order Fourier multipliers, and the acoustic–capillary phase is wave-like at low frequencies and Schrödinger-like at high frequencies. The paper is also strong in its explicit tracking of ε dependences in the dyadic dispersive estimates and in the careful treatment of non-time-integrable stationary-coefficient sources. However, the significance is conditional on closing a gap in the low-frequency Besov bound (4.19), which is used as an input for the global existence and decay arguments.","major_comments":[{"comment":"The uniform low-frequency Besov bound sup_t ||(K^{1/2}σ_ε,w_ε)(t)||_{B^{1/2}_{2,∞}} ≤ C ||(K^{1/2}σ_{ε,0},w_{ε,0})||_{B^{1/2}_{2,∞}} is asserted as a consequence of Lemmas 4.1–4.3, but the derivation is not supplied and does not follow by integrating the dyadic inequality (4.15) as written. After multiplying (4.15) by 2^{j/2} and applying Grönwall's inequality, the low-frequency contribution from j ≤ 0 contains a factor of order 2^{-j} after the damping e^{-c2^{2j}t} is integrated in time; this factor diverges as j → -∞. Lemmas 4.1 and 4.2 provide bounds of H and G only in low-regularity Besov spaces with L∞_t control, and no L^1_t integrability for the low-frequency parts of H and G is proved before (4.19). This matters because (4.19) is used to derive the decay estimates (4.21)–(4.23), which in turn feed the convergence analysis in §4.4 and the global existence argument. The gap is load-bearing for Theorem 1.2 and must be closed by a separate low-frequency argument or by replacing (4.19) with a proved estimate.","section":"§4.2, Eq. (4.19)"},{"comment":"The statement of Theorem 1.2 says that the limiting velocity u solves (1.7) with u(0) = Pu_0, but no u_0 is defined in the hypotheses. In the proof, after (4.24) the initial datum of ũ = u - u* is written as Pw_{ε,0} + (Pu*_ε - u*) = Pu_{ε,0} - u*, so the incompressible solution u is effectively taken with initial datum Pu_{ε,0} and therefore depends on ε. If a fixed limiting profile is intended, the theorem is missing an assumption such as convergence of Pu_{ε,0} to a fixed Pu_0; if the ε-dependent comparison is intended, this should be stated explicitly in the theorem rather than only appearing in the proof. This ambiguity affects the interpretation of the convergence statement (1.12), although it is likely fixable by a clarification of the statement.","section":"§1.2.2, Eq. (1.7) and §4.4, Eq. (4.24)"}],"minor_comments":[{"comment":"The operator Λ is used in the definition d_ε := Λ^{-1} div Qw_ε without having been defined in the preliminaries; it should be introduced, for example as the Fourier multiplier with symbol |ξ|.","section":"§4.3, Eq. (4.27)"},{"comment":"There are several typographical errors: “frequncy” in §2.2, “convergnce” and “defination” in §4.4. These do not affect the mathematics.","section":"§2.2 and §4.4"},{"comment":"The estimate for g_3 = κ ε² σ ∇Δσ invokes Proposition 2.2(vi) at the borderline s1 + s2 = 0 (namely s1 = -1/2 and s2 = 1/2), while the stated proposition requires s1 + s2 > 0. The authors should either justify this borderline case or use an alternative admissible product estimate, for instance by exploiting σ ∈ H^{k+3} to upgrade the low-frequency Besov regularity.","section":"Lemma 3.1, Eq. (3.8)"},{"comment":"The statement of Lemma 4.3 mentions a sequence (c_j)_{j∈Z} ∈ ℓ¹, but the displayed inequality (4.15) contains no such sequence. Either remove the reference or state explicitly which constant is summable.","section":"Lemma 4.3, Eq. (4.15)"},{"comment":"The proof of the inhomogeneous estimate (4.42) invokes Minkowski's inequality and time-translation invariance rather tersely; it should state that (4.41) is applied for each fixed τ with initial datum Φ(τ), and then the L^r_t norm is taken with respect to the forward time variable.","section":"Proposition 4.7, Eq. (4.42)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically substantial and the central mechanism is coherent, but the current version contains a load-bearing gap in the derivation of the low-frequency Besov estimate (4.19). I would be willing to reconsider after the authors supply a complete proof of that estimate or revise the argument so that the global decay does not depend on an unproved uniform low-frequency bound. The clarifying issue about the incompressible initial datum in Theorem 1.2 should also be fixed. These are within the scope of a major revision rather than grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [X],\n\nI read the Ni–Wang–Zhang–Zhang paper on the low Mach limit for the Navier–Stokes–Korteweg system with a stationary force. The headline: this is a substantial contribution, but the main non-stationary theorem has a load-bearing gap that needs to be closed before I'd trust (1.12).\n\nWhat's actually new: they construct stationary NSK solutions uniformly in ε with both density fluctuation and compressible velocity error O(ε²), improving on Deguchi's O(ε) for the non-capillary case. The stationary part (Theorem 1.1) looks solid to me; the elliptic fixed-point argument in Section 3 is coherent and the O(ε²) comes honestly from the operator B = p'(ρ∞) − κρ∞Δ. The non-stationary part introduces Korteweg-symmetric variables that make the acoustic spectral projectors uniformly bounded zero-order multipliers, and the acoustic–capillary phase is wave-like at low frequency and Schrödinger-like at high frequency. That's a genuine idea, and the dispersive estimates Lemmas 4.5–4.6 and the source decomposition in Lemma 4.10 are well thought out. The paper also avoids any fitted parameters or invented entities.\n\nThe problem is the uniform low-frequency Besov bound (4.19). It is stated as an immediate consequence of Lemmas 4.1–4.3, but no derivation is given. Lemma 4.3's dyadic inequality, after integrating and using the damping, leaves a factor 2^{-j} for j ≤ 0 that blows up as j → −∞; without a separate argument showing the low-frequency sources are time-integrable with the needed weights, the sup over j does not follow. This is not a trivial missing line: (4.19) is used to get the uniform bound (1.11), the time decay (4.23), and ultimately the convergence rate (1.12). The stress-test note about this is on target. There are also smaller issues: the global well-posedness of the limiting incompressible flow is sketched quite briefly, and the high-order estimate around (4.12) has a small index inconsistency that should be cleaned up.\n\nWho is this for? Anyone working on singular limits of capillary fluids or low-Mach limits with stationary flows. The stationary result is worth citing even now; the non-stationary result is promising but not yet established as written. My recommendation: send it to peer review. A serious referee could either close the (4.19) gap or find a counterexample. If it closes, this is a very good paper.\n\nBest.","headline":"A serious paper with a real gap: the stationary low-Mach construction is clean, but the key uniform low-frequency Besov bound (4.19) is asserted rather than proved, and it supports the main convergence theorem.","tokens_in":37776,"tokens_out":4819,"would_cite":false,"duration_ms":38706,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76N10","35B40","35B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a global-in-time low Mach number limit for the 3D compressible Navier–Stokes–Korteweg equations with a small stationary force, with an explicit convergence rate in mixed Besov norms.","keywords":["Navier-Stokes-Korteweg equations","low Mach number limit","ill-prepared initial data","stationary force","Besov spaces","dispersive estimates","acoustic-capillary phase","global strong solutions"],"falsifier":"On the linearized acoustic–capillary system around a constant state, take a high-frequency dyadic initial datum and compute its $L^r_t L^p$ norm in the regime $\\varepsilon^2\\nu_0^2\\ge 2\\rho_\\infty\\kappa$; if the norm does not decay like $\\varepsilon^{1/2-1/p}$, then the uniform projector bounds and the rate in Theorem 1.2 fail. A zero-capillarity computation, $\\kappa=0$, should show the high-frequency rate changes, confirming that the positivity condition on $l_\\varepsilon$ is load-bearing.","tokens_in":36533,"feed_emoji":"🌊","tokens_out":8818,"duration_ms":68158,"temperature":0.7,"pith_summary":"The paper proves that the three-dimensional compressible Navier–Stokes–Korteweg equations, driven by a small stationary force, admit a global-in-time low Mach number limit with an explicit convergence rate, even when the acoustic part of the initial data is not small. It first constructs stationary solutions that stay bounded uniformly in the Mach number $\\varepsilon$, with both the density fluctuation and the compressible velocity error of order $\\varepsilon^2$. For ill-prepared perturbations around these stationary flows, it establishes unique global strong solutions and shows that the density perturbation, the compressible velocity projection, and the incompressible projection error all vanish in mixed Besov norms at the rate $\\varepsilon^{\\min\\{1/r,\\,1/2-1/p\\}}$. The result matters because it shows the capillary Korteweg term improves the stationary accuracy relative to the non-capillary case and supplies a high-frequency dispersive mechanism that yields a sharp quantitative rate.","feed_headline":"Capillary compressible flow hits incompressible limit at explicit rate","feed_subtitle":"The explicit rate comes from a wave-to-Schrödinger switch in the acoustic phase.","key_machinery":"The central object is the symmetrized acoustic–capillary semigroup acting on $V=(K^{1/2}\\sigma,\\sqrt{\\rho_\\infty}d)^T$, with $K=\\gamma_0-\\kappa\\Delta$; its spectral projections satisfy uniform zero-order Fourier multiplier bounds, and its phase $\\Omega_\\varepsilon(r)=\\frac r\\varepsilon h_\\varepsilon(r)$ with $h_\\varepsilon(r)\\sim\\langle r\\rangle$ is wave-like (of size $r/\\varepsilon$) at low frequencies and Schr\\\"odinger-like (of size $r^2/\\varepsilon$) at high frequencies. Dyadic dispersive estimates convert this phase structure into the $\\varepsilon^{1/r}$ low-frequency gain and the $\\varepsilon^{1/2-1/p}$ high-frequency gain. A second mechanism is the decomposition of the Duhamel source into a time-integrable part and a stationary-coefficient part, the latter controlled by a damped estimate that does not require time integrability.","core_discovery":"The central claim is Theorem 1.2: under a small force and for data whose acoustic component is not initially small, with $2