{"id":"fbde8aa0-56a1-40c0-b192-e4649005a03c","arxiv_id":"2607.25028","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Simultaneous inviscid and low-Mach limits of 3D degenerate compressible Navier-Stokes on expanding domains yield incompressible Euler from ill-prepared data.","lead":"The paper proves that weak solutions of 3D compressible Navier-Stokes with density-dependent viscosity converge to the incompressible Euler equations when viscosity, Mach number, and domain size are sent to their singular limits together. It extends the constant-viscosity expanding-domain theory to the degenerate-viscosity case from ill-prepared data.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Theorem 2.1 quantifies over \"any weak solutions\" of (1.1), but the only existence source is an attribution to [4]; the exact match between [4]'s hypotheses and this system (boundary conditions (1.2), drag, μ(ρ)=ρ with λ=0, all γ>1) is the load-bearing point, and it is not verified in the paper.","rationale":"The reader's weakest_assumption is the same point I arrive at independently: the convergence theorem is conditional on an existence theory that is outsourced via a one-sentence attribution, and the precise hypothesis match is unchecked. I sharpen it by listing the four concrete mismatch candidates (boundary conditions, drag, BD pair, γ-range) and by noting that no uniformity in M is needed — existence per fixed (ε, M) suffices for the proof, so the concern is purely whether [4]'s theorem statement covers this exact system, not any quantitative issue. I deliberately do not treat the internal estimate blemishes as load-bearing: the (3.13) dropped-ρ slip cancels in the final identity, the Strichartz exponents in (4.19)–(4.20) are misstated but the limits still vanish, and the omitted E(0)→0 estimate is a routine computation from the stated data convergence (1.4). The reader's ACCEPT with MODERATE confidence already prices in exactly this outsourced-existence caveat, and my check confirms rather than deepens the risk, so I recommend no verdict change. If the concrete test reveals that [4] does not cover the stated boundary conditions or full γ-range, the appropriate response would be a scoped restatement of Theorem 2.1 rather than rejection, since the relative-energy core of the paper is unaffected.","tokens_in":16341,"tokens_out":10677,"duration_ms":365761,"concrete_test":"Check [4]'s main existence theorem line-by-line against Definition 2.1 and (1.1)–(1.2): does it deliver, on bounded simply connected C² domains, global weak solutions with μ(ρ)=ρ, λ=0, drag coefficient arbitrary r₁>0, boundary conditions ρu=0 and ∇ρ×n=0, energy inequality (2.4) including r₁∫ρ|u|³, for every γ>1 and for initial data of the form (1.3) (which approach vacuum-free but only L²-perturbed states)? If [4] imposes extra restrictions (e.g., on γ, on initial density bounds, or different boundary conditions), restate Theorem 2.1's hypotheses accordingly and confirm the ill-prepared data (1.3)–(1.4) remain admissible; if a mismatch is found that cannot be bridged by [6] or Mellet–Vasseur-type results, the theorem's range of applicability shrinks correspondingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the argument as a standard, competently executed relative-energy/weak-to-strong scheme: relative entropy inequality (3.2)–(3.3), acoustic filtering via Strichartz (2.12), boundary corrector w_M following [15], term-by-term absorption, Gronwall. Internally I found only repairable sloppiness, not structural gaps: (i) (3.13)'s intermediate step drops a factor ρ, but the net identity ρ∇(H''(1)(r−1))·(U−u) = −ε²ρ∂t∇Φ·(U−u) is correct via H''(1)=p'(1)=γ and the acoustic equation; (ii) decay exponents claimed in (4.19)–(4.20) for ∥∇²Φ∥_{L²} and L^{2γ/(γ−1)} (namely (1+t/ε)^{-1}) do not match (2.12) (true exponents 0 and −1/γ), but the conclusions hold anyway since ε^α·T→0 and ε^{m+1}ln(1+T/ε)→0; (iii) E(0)→0 is never shown, though it follows standardly from u_{0,ε}→u_0, ρ^{(1)}_{0,ε}→ρ^{(1)}_0 = s(0), U(0)=u_0+w_M(0); (iv) \"(m=1−1/p>2)\" in (4.9) is an evident typo. The genuinely load-bearing point is the one the reader flagged: existence. Remark 2.1 asserts that [4] (Bresch–Desjardins–Gérard-Varet 2007) yields weak solutions in the sense of Definition 2.1 \"for any data that satisfy (2.1) initially.\" But [4] is a bounded-domain result with its own structural restrictions, and the paper does not verify that [4]'s theorem literally covers: the Navier-type conditions (1.2) (ρu=0 together with ∇ρ×n=0), the quadratic drag r₁(ε)ρ|u|u for arbitrary r₁(ε)>0, the specific BD pair μ(ρ)=ρ, λ=0, and the full range γ>1 — including large γ, where the energy bound (2.6) and the min{2,γ} convergence in Theorem 2.1 are used. If [4] covers only a restricted subclass, Theorem 2.1 is vacuous (or has no objects) outside it. This is a scope/vacuity risk, not an internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper proves a simultaneous triple singular limit — vanishing viscosity (ε^α), low Mach number (ε), and expanding domains Ω_M exhausting ℝ³ — for the 3D isentropic compressible Navier–Stokes system with degenerate shear viscosity μ(ρ)=ρ, zero bulk viscosity, and a quadratic drag term. Starting from ill-prepared, compactly supported initial data, the authors show that any weak solution of (1.1) converges locally strongly (√ρu→v in L²_tL²_x(K), ρ→1) to the smooth solution of the incompressible Euler system on ℝ³, on any time interval where the latter exists. The proof is a relative-energy/weak-to-strong argument: a relative entropy inequality (Lemma 3.1) is derived from the weak formulation, the acoustic component is filtered by Strichartz estimates (2.12), the boundary corrector w_M of Feireisl–Nečasová–Sun [15] handles the expanding-domain geometry, and the remainder terms R1–R7 are absorbed via energy bounds (2.5)–(2.7) and a Cauchy–Schwarz treatment of the viscous term that avoids the Bresch–Desjardins structure, following Bisconti–Caggio–Dell'Oro [2]. The derivation is standard in strategy and, on my reading, essentially correct in execution; I found no structural gap in the convergence argument.","tokens_in":16854,"tokens_out":2597,"duration_ms":97622,"significance":"If the result stands, it is the first simultaneous inviscid + low-Mach limit for density-dependent (Bresch–Desjardins) viscosity, extending Feireisl–Nečasová–Sun (Nonlinearity 2014) from constant to degenerate viscosity, and it does so by a purely energetic argument avoiding the BD structure — a methodological simplification of independent interest. The strengths are concrete: the convergence statement is quantitative and conditional only on the existence of the target Euler solution, the proof is self-contained given the cited Strichartz estimates, and the treatment of the viscous remainder (§4, R6) is the genuinely new technical content. The main caveat is that the theorem's hypotheses quantify over weak solutions whose existence is imported, unverified, from [4]; if that import fails for the stated boundary conditions and γ-range, the theorem has no objects. This does not affect the convergence argument itself, which is sound.","major_comments":[{"comment":"Theorem 2.1 quantifies over 'any weak solutions' of (1.1), but the only source of existence is Remark 2.1, which asserts that [4] (Bresch–Desjardins–Gérard-Varet, JMPA 2007) provides weak solutions in the sense of Definition 2.1 'for any data that satisfy (2.1) initially'. This is load-bearing: if [4]'s hypotheses do not literally cover the present system, the theorem quantifies over an empty class. The manuscript does not verify the match on any of the following points: (i) the boundary conditions — [4] treats Navier-type conditions on the velocity together with a condition on ∇ρ, whereas Definition 2.1 requires ρu=0 on ∂Ω_M (a much stronger condition than Navier slip) jointly with ∇ρ×n=0; (ii) the quadratic drag r₁(ε)ρ|u|u with arbitrary coefficient r₁(ε)>0; (iii) the full range γ>1, including large γ, where [4]'s construction has documented restrictions; (iv) the pressure law p(ρ)=ρ^γ","section":"Remark 2.1 / Theorem 2.1"},{"comment":"The Gronwall conclusion at the end of §4 requires E(0)→0 as ε→0, but this is never verified in the manuscript. It follows standardly: u_{0,ε}→u_0, ρ^{(1)}_{0,ε}→ρ^{(1)}_0=s(0), and U(0)=u_0+w_M(0) with w_M(0) small by (2.13), so E(0)→0; and the pressure part of E(0) is controlled by (2.7) at t=0. However, since the whole proof is one Gronwall step on E(τ)−E(0), the initial-layer argument should be written out in a few lines rather than left implicit.","section":"§4, final paragraph"},{"comment":"The decay rates asserted for ∥∇²_xΦ∥ in L², L⁴ and L^{2γ/(γ−1)} as (1+t/ε)^{−1} do not follow from (2.12): for the L² norm (2.12) gives no decay (exponent 0), for L^{2γ/(γ−1)} it gives exponent −1/γ, and for L⁴ it gives −1/2. The final conclusion R₆₂₁≤ε^{m+1}ln(1+τ/ε)→0 is nevertheless correct, since ∫₀^τ(1+t/ε)^{−β}dt≤ε/(β−1) for β>1 and the worst case (L² part, exponent 0) is directly bounded by ε^α·T via the conservation law (2.11). But the displayed claim is wrong as written and should be corrected, with the three pieces of (4.19) estimated separately according to their actual exponents.","section":"§4, Eqs. (4.19)–(4.20)"},{"comment":"In the second equality, ρ∇_x(H''(1)s)·(U−u) is replaced by ∇_x(p'(1)s)·(U−u), silently dropping the factor ρ; the factor reappears in the final term −ε²ρ∂_t∇_xΦ·(U−u). The net identity is correct (H''(1)=p'(1)=γ and the acoustic equation (2.10)), but the intermediate line as printed is a scalar identity that does not hold; the ρ should be carried through or the step reordered.","section":"§3, Eq. (3.13)"}],"minor_comments":[{"comment":"'(m=1−1/p>2)' is an evident typo; presumably m=1−1/p with p>3, or the intended exponent should be restated.","section":"§4, Eq. (4.9)"},{"comment":"The bound |R₇|≤η(ε)→0 is asserted in one line. It follows from r₁(ε)→0 together with sup_t∥√ρu∥_{L²}≤C (2.5) and U∈L∞_tL∞_x, but the estimate should be displayed since the drag is a nonstandard feature of the model.","section":"§4, Eq. (4.16)"},{"comment":"'From to the compact support assumption on the initial data of (2.15)' — typo ('From to'), and the reference to (2.15) should be to (2.14)–(2.15) or to the initial data of (2.10).","section":"§2.4, Remark 2.2"},{"comment":"In (3.8) and elsewhere, time integrals appear inside spatial integrals written with dx only (e.g. ε^α∫∫√ρS_μ:∇_xU dxdt nested inside a dx expression); the notation should be cleaned up. Also 'We say that (ρ,u) is a weak solution ... if the following conditions are satisfied' — the second bullet (boundary conditions) lacks a bullet marker in the definition and (2.1) conflates regularity and trace conditions.","section":"§3, Eq. (3.3)/(3.8)"},{"comment":"The summation runs over R₁,…,R₇ but the viscous term is labelled R₆ᵢ (i=1,2,3) in (4.17); there is no explicitly named R₆ in (4.1). Please make the numbering consistent.","section":"§4, Eq. (4.1)"},{"comment":"Reference [2] is listed as 'J. Math. Fluid Mech. 28:46 (2026), .' with a trailing comma and no DOI; reference [23] has a typo ('applicatons'); the 2010 MSC should be updated to 2020 MSC.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a competent extension of Feireisl–Nečasová–Sun [15] and fits the journal's scope. One senior co-author has written a survey [25] on precisely the method used; the paper's novelty is the combination with the degenerate-viscosity pure-relative-energy argument of [2], which is modest but genuine. My one substantive reservation (Major comment 1) concerns the reliance on [4] for existence; if the editor can solicit the authors' verification of that point, I would be satisfied."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a clean weak-to-strong singular-limits paper: simultaneous vanishing viscosity, low Mach, and expanding domains for 3D degenerate compressible NS (viscosity ~ ρ), ill-prepared data, target incompressible Euler. That combination is new relative to Feireisl–Nečasová–Sun (constant viscosity, expanding domains) and to the recent degenerate-viscosity work (Bisconti–Caggio vanishing viscosity at fixed Mach; Fanelli–Zatorska / Chaudhuri et al. low-Mach at fixed positive viscosity). The abstract’s “first simultaneous inviscid+incompressible for density-dependent viscosity” claim checks out against the citations they give.\n\nWhat they do well is stick to a pure relative-energy argument without leaning on Bresch–Desjardins structure for the limit itself. Lemma 3.1 is the usual derivation; the remainder R1–R7 is controlled with energy bounds, Strichartz acoustics, the w_M cut-off from [15], and Cauchy–Schwarz absorption of the viscous term—the part that actually differs from constant viscosity. The expanding-domain setup correctly sidesteps boundary layers and acoustic reflection. Proof pattern is standard and, on a careful read, holds: a few exponent slips and a dropped ρ in an intermediate line of (3.13) are repairable and do not break the conclusions.\n\nThe soft spot is real but proportional: Theorem 2.1 quantifies over “any weak solutions,” and existence is imported from Bresch–Desjardins–Gérard-Varet [4] plus a vanishing drag term. The paper asserts the match (Navier-type BCs, μ=ρ, λ=0, drag, γ>1) in Remark 2.1 without checking [4] line-by-line against this exact setup. If [4] only covers a subclass, the theorem is empty outside it. That is a scope/vacuity risk, not an internal contradiction, and they flag the dependence openly. E(0)→0 is left implicit; minor.\n\nWho it’s for: people already in the Feireisl-school singular-limits / relative-energy lane. Not a paradigm shift; a competent, citable extension. Math and citation pattern look solid. I would send it to referees—existence match is exactly what a specialist referee should pin down—and I would cite it if I were writing on degenerate NS limits. Bring to reading group only if the group is deep in compressible singular limits; otherwise skip.","headline":"Solid triple-limit extension for density-dependent viscosity; the math is standard relative-energy work with one real scope caveat on existence.","tokens_in":17233,"tokens_out":608,"would_cite":true,"duration_ms":18229,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35Q31","35Q86","76N10"],"pacs":[],"model":"grok-4.5","headline":"Degenerate compressible Navier-Stokes on expanding domains converge, under simultaneous vanishing viscosity and Mach number, to incompressible Euler from ill-prepared data.","keywords":["degenerate compressible Navier-Stokes","singular limits","low Mach number","inviscid limit","expanding domains","relative energy","density-dependent viscosity","incompressible Euler"],"falsifier":"Produce a family of weak solutions satisfying the paper’s structural hypotheses on expanding domains whose density or momentum fails to converge strongly to the Euler solution on some fixed compact set inside a time interval where the smooth Euler solution still exists.","tokens_in":16732,"feed_emoji":"🌊","tokens_out":874,"duration_ms":30133,"temperature":0.7,"pith_summary":"This paper shows that three singular limits can be taken at once for three-dimensional compressible flow whose viscosity vanishes with density: viscosity to zero, Mach number to zero, and the spatial domain expanding to fill all of space. Starting from rough, ill-prepared initial data, weak solutions of the degenerate Navier-Stokes system converge locally to a smooth solution of the incompressible Euler equations for as long as that Euler solution exists. The result extends earlier work that treated only constant viscosity, and it is the first simultaneous inviscid-plus-incompressible limit for density-dependent viscosity. A sympathetic reader cares because real fluids often have viscosity that depends on density and sit in large domains; the theorem says the ideal incompressible Euler model still captures the local interior dynamics once those parameters are small.","feed_headline":"Density-dependent fluids limit to Euler as viscosity and Mach vanish","feed_subtitle":"Expanding domains let both singular limits hold at once from ill-prepared data","key_machinery":"A relative energy inequality comparing the weak Navier-Stokes solution to a corrected test field built from the Euler solution, the acoustic potential, and a boundary cut-off. The inequality absorbs acoustic oscillations, residual vacuum regions, and the degenerate viscous term so that Gronwall closes the convergence.","core_discovery":"For weak solutions of the three-dimensional degenerate compressible Navier-Stokes equations with density-dependent viscosity on a family of expanding domains, the simultaneous inviscid and low-Mach limits yield strong local convergence of density to 1 and of momentum to a smooth incompressible Euler velocity, even from ill-prepared compactly supported initial data, on any time interval short of the Euler lifespan.","pith_inferences":["The same corrector-plus-relative-energy pattern should apply verbatim on the whole space once existence without drag is available, removing the artificial friction term entirely.","Because the viscous remainder is controlled without Bresch–Desjardins entropy, the method is likely portable to other degenerate or non-Newtonian stress tensors that still admit a relative-energy inequality.","Quantitative rates could be read off from the explicit decay of the acoustic Strichartz norms and the cut-off errors once the Gronwall constant is tracked."],"forward_implications":["Local interior dynamics of slightly viscous, slightly compressible fluids with density-dependent viscosity are independent of distant boundaries once the domain is large enough.","The incompressible Euler system remains the correct target even when viscosity degenerates at vacuum, so the same ideal model covers a wider class of constitutive laws.","Ill-prepared data are admissible: acoustic waves disperse and do not prevent the low-Mach limit on expanding domains.","Any future existence theory for degenerate Navier-Stokes without drag immediately upgrades, under the same relative-energy argument, to the same triple limit."],"fun_headline_variants":["Density-dependent NS yield Euler in simultaneous inviscid-Mach limit","Expanding domains: degenerate NS converge to incompressible Euler","Ill-prepared data still give Euler from density-dependent viscosity","Inviscid low-Mach limit of degenerate NS on expanding domains","Weak solutions of density-dependent NS limit to smooth Euler"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument takes as given the existence of the weak solutions it starts from; that existence is imported from earlier theory that needs a specific viscosity law and a drag term that itself must vanish in the limit.","fun_headline_variants_meta":{"raw":{"variants":["Density-dependent NS yield Euler in simultaneous inviscid-Mach limit","Expanding domains: degenerate NS converge to incompressible Euler","Ill-prepared data still give Euler from density-dependent viscosity","Inviscid low-Mach limit of degenerate NS on expanding domains","Weak solutions of density-dependent NS limit to smooth Euler"]},"model":"grok-4.5","effort":"low","cost_usd":0.002914,"raw_usage":{"total_tokens":942,"prompt_tokens":630,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":29144000,"prompt_tokens_details":{"text_tokens":630,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":242,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":630,"tokens_out":70,"duration_ms":4463,"temperature":1.0,"reasoning_tokens":242,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T03:03:05.234235+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Produce a family of weak solutions satisfying the paper’s structural hypotheses on expanding domains whose density or momentum fails to converge strongly to the Euler solution on some fixed compact set inside a time interval where the smooth Euler solution still exists.","supporting_citations":[],"review_version":1}