{"id":"0ebddf13-e5a9-4b2a-81a5-d7a7eed6a65f","arxiv_id":"2504.18523","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For 2D vortex-sheet flows, viscous energy dissipation is shown to vanish as viscosity goes to zero, with an explicit rate in a broad setting.","lead":"This math paper studies fluids whose initial swirling motion is concentrated into a thin sheet, a 'vortex sheet.' It proves that the energy dissipated by viscosity still vanishes as viscosity tends to zero, even when external forces act, and it finds how quickly the dissipation disappears.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Initial-time layer in Theorem 5.2 depends on [11, Lemma 3.5] (L² right-continuity at t=0), which is cited but neither stated nor proved; if that lemma needs hypotheses beyond H(a)-H(c)+(5.3), the central claim is not closed.","rationale":"The paper's main construction is largely self-contained: Lemmas 2.1-2.2, Propositions 3.1-3.2 and 4.1-4.2, and the body estimate (5.13)-(5.16) are internally argued, and the new Nash refinement is a genuine quantitative contribution. The proof only needs one external load-bearing input at the initial layer: the right-continuity of the inviscid limit in L^2 at t=0, cited as [11, Lemma 3.5]. That is exactly the condition that lets the energy gap in (5.17) vanish as δ→0, so without it the central theorem is not closed even in the p>1 case. Because [11] is a preprint by two of the present authors and the lemma is not stated, a reader cannot currently certify this step. The independent De Rosa-Park result corroborates the qualitative conclusion but not the explicit rate, so it does not replace the missing lemma. Secondary gaps, notably the omitted proof of Corollary 6.7 and the reliance on [11, Proposition 4.5] in Theorem 6.1, are also present, but they are less central than the initial-layer lemma. The concern is real but addressable: if the lemma can be proved from H(a)-H(c)+(5.3) or its proof included, the argument closes. Hence the reader's CONDITIONAL verdict is appropriate, and I recommend no change.","tokens_in":16429,"tokens_out":28877,"duration_ms":292015,"concrete_test":"Obtain [11] and write out Lemma 3.5's hypotheses and proof; verify whether they are satisfied under exactly H(a)-H(c)+(5.3). In particular, check whether the proof uses a uniform L^p vorticity bound for p>1, forcing in L^∞_t L^2 rather than only L^2_t L^2, or a prior strong-convergence/energy-balance assertion. If none of these appear, the initial-layer estimate (5.17) is justified and the concern is resolved; if any does, Theorems 5.2 and 6.4 need an added hypothesis or a self-contained proof of the lemma. As a complementary numerical check, for a sequence of approximate vortex sheets with non-atomic measure initial data satisfying H(a), compute ||u(δ)−u0||_{L^2} and ν∫_0^δ ||ω^ν||^2 dt for δ,ν→0; a nonzero limit of the former as δ→0 would indicate that (5.17) does not close.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5.2 splits the dissipation into a body on [δ,T], estimated by (5.15) using the new Nash refinement, and an initial layer on [0,δ]. Equation (5.17) gives limsup_{ν→0+} ν∫_0^δ ||ω^ν||^2 dt ≤ 1/2(||u0||^2−||u(δ)||^2)+C√δ. Letting δ→0+ requires the limiting weak Euler solution u to satisfy lim_{t→0+} ||u(t)−u0||_{L^2}=0. The only justification is a citation to Lemma 3.5 of the preprint [11], whose hypotheses and proof are not reproduced. This is load-bearing because without right-continuity the energy gap in (5.17) need not vanish, and no other argument in the paper controls the layer [0,δ]. The same gap affects Theorem 6.4 and the rate (6.4), so even the p>1 vortex-sheet result is exposed. Secondary omissions, notably the unproved Corollary 6.7 and the import of [11, Proposition 4.5] in Theorem 6.1, reinforce the need to state the borrowed lemmas, but the right-continuity lemma is the single point where the main argument depends on an external condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves absence of anomalous dissipation for families of 2D Navier-Stokes solutions on the torus whose initial vorticities are nonnegative bounded Radon measures plus an L^p function, with L^2 initial velocity, allowing external forcing. The main analytic tool is a new refinement of Nash's inequality: for families of mean-free H^1 functions with uniform L^1 bound and uniformly vanishing mass in small balls, a convex increasing superquadratic function Y satisfies Y(||f||_{L^2}^2) <= ||∇f||_{L^2}^2. A measure-valued version gives the explicit growth Y(x) ∼ x|log x|^{-1/4}. These estimates are fed into the vorticity equation to show limsup_{ν→0+} ν∫_0^T ||ω^ν||_{L^2}^2 dt = 0 under a no-Diracs condition on the vorticity family, and the hypotheses are then verified for vortex-sheet initial data in BM^+ + L^p, p>1, with an explicit |log ν|^{-1/4} rate; the L^1 case is stated as Corollary 6.7. The paper also discusses an extension to time-averaged no-Diracs conditions and a counterexample illustrating the role of the hypotheses.","tokens_in":16738,"tokens_out":5788,"duration_ms":60718,"significance":"If the proof is completed as indicated, this is a substantial contribution: it extends the recent no-anomalous-dissipation result of De Rosa and Park to forced flows, allows measure-plus-L^p initial vorticities, and provides a quantitative vanishing rate. The refined Nash inequality in Propositions 3.2 and 4.2 is a clean and potentially reusable real-analysis result independent of the fluid application. The paper is honest about the role of the no-Diracs condition and gives a concrete example showing that both strong convergence of initial data and the no-Diracs condition are needed. The main weakness is that several load-bearing statements are cited from the unpublished preprint [11] without statement or proof, in one case closing the initial-time layer of the central theorem.","major_comments":[{"comment":"The concern raised about the initial-time layer is justified. After deriving (5.17), the paper closes the limit δ→0+ by invoking [11, Lemma 3.5] for L^2 right-continuity of the limiting Euler solution at t=0. This lemma is load-bearing: without it, the term (1/2)(||u_0||^2 - ||u(δ)||^2) in (5.17) need not vanish, and no other argument in the paper controls ∫_0^δ ||ω^ν||^2 dt. Since [11] is a preprint by two of the present authors and the lemma is neither stated nor proved here, the main theorem is conditional on an unverified external assumption. The authors should state the lemma with precise hypotheses and either prove it in an appendix or replace it with a published reference that covers exactly the present hypotheses.","section":"§5, proof of Theorem 5.2, Eq. (5.17)"},{"comment":"The abstract advertises the main result for 'an arbitrary non-negative measure plus an integrable function' as initial vorticity, which is exactly the content of Corollary 6.7 with H3(a)-H3(b). However, the proof is omitted as 'standard'. Since this corollary is one of the paper's advertised results and depends on the same initial-layer argument as Theorem 5.2, the omission is a gap in the manuscript's central claim. The proof should be included, or the abstract and corollary should be explicitly downgraded to a conjecture/sketch.","section":"§6, Corollary 6.7"},{"comment":"The verification of the no-Diracs condition (5.3) relies on the maximal-function estimate (6.1), quoted from [11, Proposition 4.5] without proof. This proposition is used to propagate uniform integrability of the initial and forcing vorticities to all times, so it is essential for Theorem 6.1 to imply H(c) and (5.3). Like Lemma 3.5, this should be stated explicitly and either proved or given a fully verifiable published reference, since [11] is not yet part of the public peer-reviewed record.","section":"§6, proof of Theorem 6.1, Eq. (6.1)"}],"minor_comments":[{"comment":"The sentence citing [11, Lemma 3.5] gives no indication of the lemma's hypotheses; even a brief statement of the exact assumptions would help the reader see that they match H(a)-H(c) and (5.3).","section":"§5, text after (5.17)"},{"comment":"In the line after (4.4), the expression '||µ||_{H^{-1}}| + ||w||_{L^p}' contains a stray vertical bar after the H^{-1} norm; this is likely a typographical error.","section":"§4, Proposition 4.1 proof"},{"comment":"The notation '4√log' in (4.2), (4.6), (4.7) is ambiguous; writing (log x)^{1/4} or \\sqrt[4]{\\log x} would make the estimates easier to read.","section":"§3-§4, notation"},{"comment":"The statement uses N ∈ R^+ but then conflates N with the integer ⌈N⌉ in the proof; a remark clarifying that all estimates are unchanged up to constants would remove the ambiguity.","section":"§2, Lemma 2.2"},{"comment":"The claimed generalizations of Propositions 3.1, 3.2, 4.1, and 4.2 to time-averaged conditions (B') and (B'') are only sketched. Since these are presented as extensions rather than as main theorems, a sketch may be acceptable, but the definitions of the analogous η and Φ functions should at least be written down.","section":"§7, extension discussion"},{"comment":"In the example, the notation '~Cδ0' appears instead of a properly typeset multiple of a Dirac delta; this should be corrected.","section":"§7, example"}],"recommendation":"major_revision","confidential_remarks":"The key issue for the editor is the reliance on the unpublished preprint [11], which has two of the present authors as co-authors. This is not a novelty concern but a verifiability concern: the referee cannot check whether [11, Lemma 3.5] holds under the exact hypotheses of this paper. I would request that the authors include the statement and proof of the cited lemma, and also of [11, Proposition 4.5], or restrict the scope of their theorems accordingly. The independent overlap with De Rosa--Park is acknowledged in the paper and does not, by itself, affect my recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take: this is a real result, with a genuinely new technical tool, but the qualitative conclusion was already out there. De Rosa and Park proved no anomalous dissipation for vortex sheets; Elgindi et al. prove it again, with forcing, more flexible approximation, and an explicit |log nu|^{-1/4} rate for the measure+L^p case. They are upfront about the overlap. The new refined Nash inequality (Prop 3.2 and its measure-valued version Prop 4.2) is the real contribution, and it looks correct to me.\n\nThe proof is mostly careful and readable. Lemmas 2.1-2.2 and the Nash refinements are worked out in detail. The structure of Theorem 5.2 is sound: split the dissipation into [delta,T] and [0,delta]; on the body, the refined inequality plus Jensen gives the decay; on the initial layer, the energy equation and weak lower-semicontinuity reduce things to the energy gap of the limit.\n\nThe soft spot is exactly the initial layer. Equation (5.17) leaves a term involving ||u0||^2 - ||u(delta)||^2, and the proof lets delta go to 0 using a cited lemma [11, Lemma 3.5] that the inviscid weak solution is right-continuous in L^2 at t=0. The lemma is not stated, and its hypotheses are not checked in the text. This is not a manufactured flaw: the claim is load-bearing. Without right-continuity, the initial layer might not vanish, and then the theorem does not close. I suspect the lemma is true under the given hypotheses, but I could not verify it from this paper. The authors should either prove it or state it precisely and verify its hypotheses. The same goes for the use of [11, Prop 4.5] in Theorem 6.1, and Corollary 6.7 is omitted as 'standard'.\n\nMinor: the abstract says 'provides an explicit estimate for the dissipation'; the explicit rate appears only in Theorem 6.4. In Theorem 5.2 the rate is non-explicit. Slightly soften that.\n\nThis is for people working on vanishing viscosity and 2D turbulence; the inequality itself is reusable. Bottom line: it deserves a serious referee. The analytic core is new and the conclusion is important, even if not first. I would accept for review, with a referee asked to push for a self-contained treatment of the initial-time layer and the borrowed lemmas.","headline":"A technically solid and genuinely new quantitative proof of no anomalous dissipation for vortex sheets, but the initial-time layer leans on an unstated borrowed lemma that the referee should check.","tokens_in":17269,"tokens_out":2682,"would_cite":true,"duration_ms":26744,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35Q31","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Viscous energy loss vanishes for 2D vortex-sheet flows.","keywords":["anomalous dissipation","vortex sheets","Navier-Stokes","vanishing viscosity","Nash inequality","vorticity","energy dissipation","2D Euler"],"falsifier":"A decisive observation would be a family satisfying hypotheses H(a)–H(b) and H2(a)–H2(b) for which limsup_{ν→0⁺} ν∫₀ᵀ‖ω^ν‖²_{L²}dt > 0, or alternatively a demonstration that the L² right-continuity lemma fails under the stated hypotheses. The Section 7 Dirac-type example shows how the conclusion fails when the no-Diracs and strong-initial-data conditions are dropped, but it does not violate the hypotheses of the theorems, so it does not falsify them.","tokens_in":16235,"feed_emoji":"🌊","tokens_out":5956,"duration_ms":56550,"temperature":0.7,"pith_summary":"This paper proves that families of two-dimensional Navier-Stokes solutions with vortex-sheet initial data do not exhibit anomalous dissipation: the viscous dissipation term ν∫₀ᵀ‖ω^ν‖²_{L²}dt tends to 0 as ν→0⁺. The initial vorticity may be any nonnegative bounded Radon measure plus an Lᵖ function (or L¹), the initial velocity is only in L², and an external force is allowed. The proof is quantitative and yields an explicit rate, of order |log ν|^{−1/4}, in the vortex-sheet case. A sympathetic reader should care because anomalous dissipation is a hallmark of turbulent energy loss in the inviscid limit, and this removes it for a broad class of singular planar flows.","feed_headline":"Viscous energy loss vanishes for 2D vortex sheets","feed_subtitle":"A refined Nash inequality gives a |log ν|^{−1/4} vanishing rate even with forcing.","key_machinery":"The load-bearing object is a refinement of Nash's inequality in dimension two, stated as Proposition 3.2 and extended to bounded Radon measures in Proposition 4.2. For a family of mean-free H¹ functions with uniformly bounded L¹ norm and uniformly vanishing mass in small disks, the proposition produces a convex, increasing, superquadratic function Υ such that Υ(‖f‖²_{L²}) ≤ ‖∇f‖²_{L²}; when f is a nonnegative measure plus an Lᵖ function, Υ can be chosen so that its inverse-square-root behaves like x|log x|^{−1/4} at infinity. Applied to the vorticity ω^ν(·,t) at each time, this inequality converts the known global bound on ‖∇ω^ν‖²_{L²} into a lower bound on ν²∫Υ(‖ω^ν‖²)dt, and Jensen's inequality turns that into the claimed vanishing rate for the dissipation.","core_discovery":"The central claim is Theorem 6.4 together with Remark 6.6: if the initial vorticities decompose as μ^ν + w^ν with μ^ν ≥ 0 bounded in BM, w^ν bounded in Lᵖ for some p > 1, and the curls of the forcings are bounded in L¹(0,T;Lᵖ), then every physically realizable weak limit satisfies limsup_{ν→0⁺} ν∫₀ᵀ‖ω^ν(·,t)‖²_{L²}dt = 0, with the dissipation on any interval [δ,T] bounded by a constant times |log ν|^{−1/4}. A more general theorem (Theorem 5.2) proves the same vanishing conclusion for any vanishing-viscosity family satisfying hypotheses H(a)–H(c) plus the no-Diracs condition that vorticity mass in small disks tends to zero uniformly. The authors also show the method extends to a time-averaged version of the no-Diracs condition and yields the same rate for the singular part. This is a statement about absence of anomalous dissipation only; it does not assert strong convergence of the velocity approximations.","pith_inferences":["The logarithmic rate suggests the mechanism is robust: any singularity class whose small-scale mass decays at least logarithmically may yield a vanishing rate, so the method could extend to other critical Besov or Morrey-type initial data.","The time-averaged no-Diracs extension in Section 7 implies the result should be stable under oscillations in time of the vortex sheet, not just spatial concentration.","A natural test is whether the |log ν|^{−1/4} rate is sharp; constructing an example that saturates the bound would reveal whether the refined Nash inequality is lossless.","Combined with the equivalence proved in the companion preprint [11], absence of anomalous dissipation is necessary but not sufficient for energy balance of the inviscid limit; the missing ingredient is strong L² convergence, which this method does not attempt to provide."],"forward_implications":["Dissipation vanishes for vortex-sheet initial data: ν∫₀ᵀ‖ω^ν‖²_{L²}dt → 0 as ν→0⁺, even though the velocity family need not converge strongly in L².","The rate |log ν|^{−1/4} quantifies how quickly the dissipation disappears; the paper does not claim this rate is optimal.","Forced flows are covered, so the result applies beyond the unforced Navier-Stokes setting.","Absence of anomalous dissipation is compatible with the limiting weak solution carrying an energy defect; it is strictly weaker than strong convergence.","The Section 7 example shows the conclusion can fail when both the no-Diracs and strong-initial-data conditions are violated, so the hypotheses are not vacuous."],"supporting_citations":[{"why":"Supplies the L² right-continuity of the inviscid solution at t=0 (Lemma 3.5) used to close the initial time layer, and the maximal-function estimate (6.1) that propagates uniform integrability.","marker":"[11]"},{"why":"The classical Nash inequality that Proposition 3.2 refines; it is the starting point for the new quantitative estimate.","marker":"[17]"},{"why":"Establishes existence of weak solutions for vortex sheets of distinguished sign, adapted here to show that the weak limit of the viscous approximations is an Euler weak solution.","marker":"[4]"},{"why":"Provides the mass-in-small-disks estimate (4.1) for nonnegative measures in H⁻¹, used in the measure-valued extension of the refined inequality.","marker":"[20]"},{"why":"Independent proof of no anomalous dissipation for this class; the present work differs by covering forcing and giving an explicit vanishing rate.","marker":"[18]"},{"why":"Existence of weak solutions for L¹∩H⁻¹ initial vorticity, adapted in Remark 6.2 and Theorem 6.1 to produce approximating families satisfying the hypotheses.","marker":"[21]"},{"why":"Remarks on weak solutions for vortex sheets with distinguished sign, used in Remark 6.5 to justify that mollified initial data yield physically realizable solutions.","marker":"[16]"}],"fun_headline_variants":["Zero anomalous dissipation for vortex sheets","2D vortex sheets evade viscosity energy loss","Anomalous dissipation ruled out for sheets","Vortex sheets: no viscous energy leak at low ν","Dissipation vanishes for 2D sheet flows"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on a lemma borrowed from the companion preprint [11] that the limiting Euler solution is right-continuous in L² at t=0; if that lemma requires hypotheses stronger than those verified here, the treatment of the initial time layer [0,δ] would have a gap.","fun_headline_variants_meta":{"raw":{"variants":["Zero anomalous dissipation for vortex sheets","2D vortex sheets evade viscosity energy loss","Anomalous dissipation ruled out for sheets","Vortex sheets: no viscous energy leak at low ν","Dissipation vanishes for 2D sheet flows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00012,"raw_usage":{"total_tokens":1055,"prompt_tokens":876,"completion_tokens":179,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":111}},"tokens_in":492,"tokens_out":179,"duration_ms":2327,"temperature":1.0,"reasoning_tokens":111,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:15:05.851878+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive observation would be a family satisfying hypotheses H(a)–H(b) and H2(a)–H2(b) for which limsup_{ν→0⁺} ν∫₀ᵀ‖ω^ν‖²_{L²}dt > 0, or alternatively a demonstration that the L² right-continuity lemma fails under the stated hypotheses. The Section 7 Dirac-type example shows how the conclusion fails when the no-Diracs and strong-initial-data conditions are dropped, but it does not violate the hypotheses of the theorems, so it does not falsify them.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the L² right-continuity of the inviscid solution at t=0 (Lemma 3.5) used to close the initial time layer, and the maximal-function estimate (6.1) that propagates uniform integrability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical Nash inequality that Proposition 3.2 refines; it is the starting point for the new quantitative estimate."},{"cited_title":"Delort, Existence of vortex sheets in dimension two , J","cited_arxiv_id":null,"evidence_quote":"Establishes existence of weak solutions for vortex sheets of distinguished sign, adapted here to show that the weak limit of the viscous approximations is an Euler weak solution."},{"cited_title":"Schochet, The weak vorticity formulation of the 2-D Euler equations an d concentration- cancellation , Commun","cited_arxiv_id":null,"evidence_quote":"Provides the mass-in-small-disks estimate (4.1) for nonnegative measures in H⁻¹, used in the measure-valued extension of the refined inequality."},{"cited_title":"Vecchi and S","cited_arxiv_id":null,"evidence_quote":"Existence of weak solutions for L¹∩H⁻¹ initial vorticity, adapted in Remark 6.2 and Theorem 6.1 to produce approximating families satisfying the hypotheses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Remarks on weak solutions for vortex sheets with distinguished sign, used in Remark 6.5 to justify that mollified initial data yield physically realizable solutions."}],"review_version":1}