{"id":"8debf4a4-3bb1-4201-98ea-ee8226b4a944","arxiv_id":"2606.04218","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Local anomalous dissipation vanishes for 2D NS solutions bounded in L^{1+}_t L^∞_{x,loc} away from the boundary, yielding convergence to an Euler solution whose large-scale approximation satisfies local energy balance without pressure bounds.","lead":"The paper proves that anomalous dissipation vanishes away from boundaries in 2D Navier-Stokes flows when velocity is uniformly bounded in L^{1+}_t L^∞_{x,loc}. A smart generalist might read it to see concrete conditions under which energy is conserved in the zero-viscosity limit for flows in bounded domains.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Transfer of localization/modulation + vorticity estimates to interior may require unstated controls on pressure or boundary-induced terms","rationale":"The reader's weakest_assumption correctly isolates the single step whose validity is not guaranteed by the abstract or by the cited works alone. The full-text description does not alter this; the absence of pressure bounds is presented as a feature, yet it is precisely the quantity that would normally control the nonlocal terms arising from localization near a no-slip boundary.","tokens_in":1795,"tokens_out":399,"duration_ms":15290,"concrete_test":"In the section deriving the local dissipation estimate (likely §3 or §4), extract the identity obtained after applying the modulation/localization operator to the vorticity equation; recompute the pressure term and any integration-by-parts remainder explicitly on a test domain touching the interior but not the boundary. If either term is bounded solely by the given L^{1+}_t L^∞ norm without invoking a global pressure estimate, the claim holds; otherwise the extension fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The vanishing of local anomalous dissipation is obtained by adapting localization-via-modulation (AD23, CW23) and vorticity-energy/structure-function estimates (CFLS16, DP25dissconc) to an interior region Ω' ⋐ Ω. These source arguments are typically global or periodic; the no-slip condition on ∂Ω can generate nonlocal pressure contributions and vorticity boundary layers that propagate into Ω' even at positive distance. The stated hypothesis supplies only a local L^{1+}_t L^∞_{x,loc} bound on u^ν and no uniform pressure control. If the adaptation in the manuscript does not explicitly cancel or absorb the resulting commutator/pressure terms using only the local bound, the dissipation estimate does not close.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves absence of local anomalous dissipation for 2D Navier-Stokes with no-slip boundary conditions away from the boundary, under the assumption that u^ν is uniformly bounded in the Onsager-supercritical space L^{1+}_t L^∞_{x,loc} (with suitable initial data). The argument adapts localization-via-modulation techniques from AD23 and CW23 together with vorticity-energy and L^2 structure-function estimates from CFLS16 and DP25dissconc. It further establishes convergence to an Euler solution whose large-scale approximation (in the sense of PGLR18) satisfies a local energy balance, without any uniform-in-viscosity pressure bounds.","tokens_in":1929,"tokens_out":611,"duration_ms":19842,"significance":"If the localization closes, the result would extend inviscid-limit and anomalous-dissipation theory to bounded domains with physical boundary conditions while avoiding pressure control; the application of the large-scale approximation to the 2D inviscid limit is new. The explicit use of only local supercritical bounds is a potentially useful strengthening over global assumptions in prior works.","major_comments":[{"comment":"The central claim that local anomalous dissipation vanishes rests on transferring the localization-via-modulation and vorticity estimates to an interior subdomain Ω' ⋐ Ω. The no-slip condition on ∂Ω generates nonlocal pressure contributions and possible vorticity boundary-layer effects that reach into Ω' at positive distance; the manuscript supplies only the local L^{1+}_t L^∞_{x,loc} bound and states that no uniform pressure control is assumed. It is therefore necessary to verify, in the derivation of the localized energy balance (the step that closes the dissipation estimate), that all resulting commutator and pressure terms are absorbed using solely the given local bound. If these terms are not explicitly cancelled or estimated, the vanishing result does not follow.","section":"Proof of the vanishing of local anomalous dissipation (adaptation of AD23/CW23 and CFLS16/DP25dissconc estimates)"},{"comment":"The subsequent convergence statement to an Euler solution with local energy balance for the large-scale approximation likewise depends on the same localized dissipation control. Any unaccounted boundary-induced error in the interior estimates would propagate into the limit and undermine the local energy balance claim.","section":"Convergence to Euler and local energy balance for the large-scale approximation"}],"minor_comments":[{"comment":"The abstract contains the typographical error \"uniformily\" (should be \"uniformly\").","section":"Abstract"},{"comment":"Notation for the interior subdomain Ω' and the precise meaning of the local bound (including the precise dependence on dist(Ω',∂Ω)) should be stated explicitly at the beginning of the main theorem.","section":"Statement of main results"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the opportunity to address these points. We respond to each major comment below.","responses":[{"response":"In the derivation of the localized energy balance in Section 3, the cutoff function is chosen with support in Ω' at positive distance from ∂Ω. The modulation technique is applied within this interior region, and the commutator terms (including those arising from the pressure) are controlled via the local L^{1+}_t L^∞ bound combined with the L^2 structure-function estimates adapted from DP25dissconc. These estimates absorb the contributions without global pressure bounds because the distance to the boundary ensures that nonlocal effects remain controllable locally. The vorticity-energy estimates from CFLS16 are likewise localized to Ω', so boundary-layer influences do not enter the interior balance at the scales considered. We will add a short clarifying paragraph in the revised manuscript explicitly listing the absorption steps for these terms.","revision_made":"partial","referee_comment":"[Proof of the vanishing of local anomalous dissipation (adaptation of AD23/CW23 and CFLS16/DP25dissconc estimates)] The central claim that local anomalous dissipation vanishes rests on transferring the localization-via-modulation and vorticity estimates to an interior subdomain Ω' ⋐ Ω. The no-slip condition on ∂Ω generates nonlocal pressure contributions and possible vorticity boundary-layer effects that reach into Ω' at positive distance; the manuscript supplies only the local L^{1+}_t L^∞_{x,loc} bound and states that no uniform pressure control is assumed. It is therefore necessary to verify, in the derivation of the localized energy balance (the step that closes the dissipation estimate), that all resulting commutator and pressure terms are absorbed using solely the given local bound. If these terms are not explicitly cancelled or estimated, the vanishing result does not follow."},{"response":"The convergence argument and the local energy balance for the large-scale approximation (Section 4) are direct consequences of the vanishing of local anomalous dissipation proved in Section 3. Because the interior estimates close using only the given local bound (as outlined above), no boundary-induced errors remain to propagate into the inviscid limit. The large-scale approximation is applied on subdomains strictly inside Ω, where the uniform-in-viscosity control holds.","revision_made":"no","referee_comment":"[Convergence to Euler and local energy balance for the large-scale approximation] The subsequent convergence statement to an Euler solution with local energy balance for the large-scale approximation likewise depends on the same localized dissipation control. Any unaccounted boundary-induced error in the interior estimates would propagate into the limit and undermine the local energy balance claim."}],"tokens_in":1516,"tokens_out":579,"duration_ms":21457,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper shows that if the Navier-Stokes approximations stay bounded in L^{1+}_t L^∞ locally in the interior, then local anomalous dissipation vanishes for 2D no-slip flows, and the Euler limit has its large-scale approximation satisfying local energy balance. They reach this without any uniform pressure bounds, which is the practical point they highlight.\n\nWhat is new is the combination: localization via modulation from AD23 and CW23, vorticity and structure-function estimates from CFLS16 and DP25dissconc, plus the first application of the large-scale approximation from PGLR18 to an inviscid-limit question. The abstract presents this as a coherent chain that closes under the stated local bound and initial data assumptions.\n\nThe soft spot is exactly the one the stress-test flags. The source arguments for localization and vorticity estimates were mostly developed in periodic or whole-space settings. Moving them to an interior subdomain Ω' ⋐ Ω with a no-slip boundary outside could introduce pressure commutators or boundary-layer effects that the local velocity bound alone may not control. The paper states that the estimates work without pressure control, but whether the details actually absorb those terms is the load-bearing step that needs checking in the full text.\n\nThis is for specialists working on 2D inviscid limits, Onsager-type questions, and energy balance in bounded domains. A reader already familiar with the cited localization and large-scale tools will see the value in the new setting.\n\nIt is a precise conditional result with a clear strategy, so it deserves a serious referee even if the boundary adaptation requires some extra work in revision.","headline":"Conditional vanishing of local anomalous dissipation away from the boundary under local L^{1+}_t L^∞ bound, with first use of large-scale approximation in 2D no-slip inviscid limit, but the interior adaptation of cited estimates is the part to verify.","tokens_in":2446,"tokens_out":426,"would_cite":false,"duration_ms":19786,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Uniform bounds in Onsager supercritical space make local anomalous dissipation vanish away from the boundary in 2D Navier-Stokes.","keywords":["Navier-Stokes equations","anomalous dissipation","inviscid limit","2D incompressible flow","local energy balance","Onsager critical spaces"],"falsifier":"Finding a sequence of solutions with the stated L^{1+}_t L^∞ bound but positive local anomalous dissipation in the interior would disprove the vanishing result.","tokens_in":2668,"feed_emoji":"","tokens_out":590,"duration_ms":20287,"temperature":0.7,"pith_summary":"The paper shows that for 2D Navier-Stokes equations with no-slip boundaries, if the velocity is uniformly bounded in L^{1+}_t L^∞_{x,loc} with controlled initial data, then anomalous dissipation vanishes in the interior. This leads to convergence to an Euler solution where the large scale approximation satisfies a local energy balance. The proof uses localization via modulation and vorticity energy estimates without needing bounds on the pressure. Such a result matters because it identifies conditions under which the viscous term disappears locally in the inviscid limit for flows in bounded domains.","feed_headline":"Bounded velocity removes local dissipation anomaly in 2D interior","feed_subtitle":"L^{1+}_t L^∞ control suffices for vanishing anomalous dissipation away from no-slip walls and local energy balance in Euler limit.","key_machinery":"Localization via modulation together with vorticity energy estimates and L^2-based structure functions.","core_discovery":"We show that anomalous dissipation vanishes locally away from the boundary for solutions of the 2D Navier-Stokes equations with no-slip boundary condition if the velocity u^ν is uniformly bounded in the Onsager supercritical space L^{1+}_t L^∞_{x,loc} with appropriate initial conditions. This setting produces convergence to an Euler solution whose large scale approximation satisfies a local energy balance equation, without assuming uniform-in-viscosity bounds on the pressure.","pith_inferences":["Boundary layers are likely responsible for any persistent anomalous dissipation in 2D flows.","Similar localization techniques might apply to other dimensions or boundary conditions if adapted.","Testing the bound numerically in periodic domains or with slip boundaries could isolate interior behavior."],"forward_implications":["The inviscid limit converges to an Euler solution in the interior.","The large scale approximation of the limit satisfies local energy balance.","These conclusions hold without uniform bounds on the pressure.","The result is specific to two-dimensional incompressible flows away from boundaries."],"fun_headline_variants":["Local anomalous dissipation absent in 2D interior under velocity bounds","Bounded velocity yields no local dissipation anomaly in 2D flows","2D Euler limit satisfies local energy balance with no pressure bounds","Local energy balance for large scale Euler approximation in 2D limit"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The cited localization and vorticity estimate techniques apply directly to regions away from the no-slip boundary without change.","fun_headline_variants_meta":{"raw":{"variants":["Local anomalous dissipation absent in 2D interior under velocity bounds","Bounded velocity yields no local dissipation anomaly in 2D flows","2D Euler limit satisfies local energy balance with no pressure bounds","Local energy balance for large scale Euler approximation in 2D limit"]},"model":"grok-4.3","cost_usd":0.004461,"raw_usage":{"total_tokens":2235,"prompt_tokens":686,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":44612000,"prompt_tokens_details":{"text_tokens":686,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1478,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":686,"tokens_out":71,"duration_ms":10494,"temperature":1.0,"reasoning_tokens":1478,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T08:51:37.391335+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding a sequence of solutions with the stated L^{1+}_t L^∞ bound but positive local anomalous dissipation in the interior would disprove the vanishing result.","supporting_citations":[],"review_version":1}