{"id":"c6206a8d-d904-4a03-b5f6-4b514289a930","arxiv_id":"2412.13066","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Baire generic L^2 data, 3D Navier-Stokes weak solutions fail to belong to supercritical L^r(0,∞;L^s) spaces, and the critical Besov threshold is equivalent to a scaling-matched a priori viscosity bound.","lead":"This math paper proves that for Baire generic square-integrable initial data, 3D Navier-Stokes equations have no weak solution in L^r L^s spaces in the supercritical range, including the classical L^4(0,T;L^4) class. It also shows such integrability is equivalent to a scaling-matched a priori estimate, and gives conditional consequences for uniqueness, energy equality, and anomalous dissipation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.3, which yields the headline generic failure of L4(0,T;L4), is stated without proof and is not a direct consequence of Theorem 1.2's global-in-time statement.","rationale":"The reader's conditional accept centers on Fatou/homogeneity and weak-star closedness. I checked that part of Theorem 1.2 and found no fatal flaw: Y_M is weakly sequentially closed for sequences with bounded Z-norm, and the Fatou assumption supplies the limit in L^rX. The scaling algebra in (i)⇒(ii) is consistent. The actual weak point is Corollary 4.3. It is the only result in the paper that produces the abstract's generic nonexistence of L^4(0,T;L^4) integrability, and it is not proved. Since the corollary is a nontrivial local-in-time reduction, not a formal corollary of Theorem 1.2, the central advertised conclusion rests on an omitted argument. This is a completeness problem, not evidence of falsity; hence the verdict stays conditional rather than reject. A full proof of Corollary 4.3, or an explicit derivation from Theorem 1.2, would remove the concern.","tokens_in":29066,"tokens_out":35582,"duration_ms":327007,"concrete_test":"Write out the proof of Corollary 4.3. Specifically, define Y_{M,T} as the set of data admitting a weak solution in L^r(0,T;L^s) with norm at most M; after Lemma 4.1 obtain B(0,ε)⊂Y_{M,T}. For fixed u_0≠0 choose w_0(x)=λ^{-1}u_0(x/β) with β chosen so that w_0 lies in the ball, and track the rescaled solution u_λ(x,t)=λ w(βx,αβt) with viscosity ν_λ=(λ/β)ν_0. Verify that the L^r norm over the growing interval decays by the factor λ^{1-5/(3r)-2/s} and that the weak limit is (0,0), contradicting the nonzero initial datum. If this scaling computation cannot be completed, Corollary 4.3 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.3 states Corollary 4.3 immediately after Proposition 4.2 and then moves to Section 5; no proof is supplied. The abstract's advertised resolution of the open L^4 integrability question depends on the first alternative of Corollary 4.3, not on Theorem 1.2 itself: Theorem 1.2 concerns global membership in L^r(0,∞;X), whereas the corollary concerns the union over T>0 of L^r(0,T;L^s). To pass from a non-meagre set of data admitting local solutions to the Baire ball B(0,ε) and then to the inviscid contradiction requires a separate scaling/restart argument. The natural reduction would rescale the local solution and check that the L^r(0,T/(λβ);L^s) norm decays like λ^{1-5/(3r)-2/s}; positivity is exactly the corollary's hypothesis 5/(3r)+2/s<1. None of these steps appears in the text. Until the proof is supplied, the central claim that Baire-generic L^2 data admit no L^4(0,T;L^4) weak solution is not established by the written argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies generic integrability properties of weak and Leray--Hopf solutions of the 3D Navier--Stokes equations. Theorem 1.2 relates, for suitable $s$-homogeneous spaces $X$ and $2$-homogeneous spaces $Z$, the existence of weak solutions in $L^r(0,\\infty;X)$ for a non-meagre set of data to uniform a priori estimates, via a Baire-category argument based on a translation lemma and weak-star compactness. Corollary 4.3 then claims that, for $5/(3r)+2/s<1$, generic data admit no weak solution in $\\cup_{T>0}L^r(0,T;L^s)$, in particular ruling out $L^4(0,T;L^4)$ integrability for generic $L^2_\\sigma$ data. Further results give a Besov-space corollary, time-weighted LPS equivalences, and conditional statements on generic uniqueness and energy equality under global regularity assumptions.","tokens_in":29308,"tokens_out":25995,"duration_ms":239224,"significance":"If the central claims are correct, the paper resolves, in the Baire-generic sense, the previously open question whether Leray--Hopf solutions belong to $\\cup_{T>0}L^4(0,T;L^4)$, and it provides a clean scaling-based equivalence between existence of solutions in a given space and a priori bounds. The use of the nonlinear open-mapping/translation lemma from [40] is interesting and is developed in a self-contained way in Lemma 4.1; the Fatou lemma for Bochner--Besov spaces in Lemma 2.3 is also a useful explicit tool. However, the advertised headline result is contained in Corollary 4.3, which is stated without proof, and several load-bearing proof steps have gaps that need to be addressed before the main conclusions can be considered established.","major_comments":[{"comment":"Corollary 4.3, which is the basis for the advertised generic failure of $\\cup_{T>0}L^4(0,T;L^4)$, is stated without proof. It is not a direct consequence of Theorem 1.2: Theorem 1.2 concerns global membership in $L^r(0,\\infty;X)$, whereas the corollary concerns the union over $T>0$ of $L^r(0,T;L^s)$. Passing from a non-meagre set of data admitting local solutions to the Baire ball $B(0,\\varepsilon)$ and then to the first alternative of the corollary requires a separate scaling/restart argument, together with a check of the sign of $5/(3r)+2/s-1$; none of these steps appears in the text. Until this proof is supplied, the central claim about $L^4(0,T;L^4)$ is not established by the written argument.","section":"§4.3, Corollary 4.3"},{"comment":"The viscosity rescaling is written with the wrong parameter. For a solution $u$ of the $\\tilde\\nu$-equation, $u_\\lambda(x,t)=u(x/\\lambda,t/\\lambda)$ solves the $\\nu$-equation only when $\\lambda=\\tilde\\nu/\\nu$, not when $\\lambda=\\nu/\\tilde\\nu$ as stated; with $\\lambda=\\nu/\\tilde\\nu$ it solves an equation with viscosity $\\tilde\\nu^2/\\nu$ rather than $\\nu$. Moreover, even for the correct choice of $\\lambda$, the displayed identity $\\|u_\\lambda\\|_{L^\\infty(0,\\infty;L^2)}+\\sqrt{\\nu}\\|\\nabla u_\\lambda\\|_{L^2(0,\\infty;L^2)}=\\lambda^{3/2}(\\|u\\|_{L^\\infty(0,\\infty;L^2)}+\\sqrt{\\tilde\\nu}\\|\\nabla u\\|_{L^2(0,\\infty;L^2)})$ is not correct: the $L^2$ spatial norm scales as $\\lambda^{3/2}$, while the $\\sqrt{\\nu}\\|\\nabla u_\\lambda\\|$ term scales with a different factor. Since this step is used to derive (9)--(10), the proof of the equivalence (i)$\\Leftrightarrow$(ii) needs correction.","section":"§4.1, proof of (i)⇒(ii)"},{"comment":"The proof asserts that the sets $Y_M:=\\{u_0\\in L^2_\\sigma\\cap Z: \\exists u\\in N_\\nu(u_0),\\ \\|u\\|_{L^r(0,\\infty;\\dot B^{\\beta_r}_{r,\\infty})}\\le M\\}$ are closed, and then applies Lemma 4.1 to transfer a ball to the origin. But $N_\\nu$ denotes the Leray--Hopf class, and Proposition 2.17 together with Remark 2.18 explicitly notes that this class is not known to be weakly-star sequentially closed; Proposition 2.17 only gives closure under strong convergence for Leray--Hopf solutions. Lemma 4.1 requires closure under the weak topology, because translations converge weakly and not strongly. Therefore the centering argument in Corollary 1.3 is unjustified as written.","section":"§4.2, proof of Corollary 1.3"}],"minor_comments":[{"comment":"Remark 3.3 asserts a stronger generic nonexistence result for very weak solutions but states that the details are omitted; since this is an unproved claim, it should either be proved or explicitly marked as conditional and not part of the main results.","section":"§3, Remark 3.3"},{"comment":"In the definition of $p$-homogeneous spaces, the notation $\\approx_Z$ is used without explanation; the subscript $Z$ also collides with the later use of $Z$ as the space of initial data. Please clarify the intended meaning of the equivalence constant.","section":"Definition 1.1"},{"comment":"In the displayed inequalities of the proof of Lemma 2.3, the upper limit $T$ appears without prior definition; the statement concerns the interval $(0,\\infty)$, so the limit should be $\\infty$.","section":"Lemma 2.3, proof"},{"comment":"References [43] and [44] appear to be the same paper listed twice with identical title, journal, and page numbers; please merge or correct the entries.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper has a genuinely interesting idea and the main theorem, if repaired, would be a valuable contribution. However, the headline application in Corollary 4.3 is unproved, and the scaling step in the proof of Theorem 1.2 contains an apparent inconsistency that affects the derivation of the a priori estimates. The Leray--Hopf closedness issue in Corollary 1.3 is also directly connected to a limitation acknowledged in Remark 2.18. I would encourage a revision that supplies the missing proof of Corollary 4.3, corrects the rescaling, and clarifies the function-space closedness assumptions. Self-citation of [40] is appropriate here, since the main technique genuinely comes from that paper and Lemma 4.1 is restated and proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem of this paper is solid and genuinely useful, but the advertised headline—that Baire-generic L^2 data admit no L^4(0,T;L^4) weak solution—is not established by the written argument. The claim rests on Corollary 4.3, which is stated without proof and does not follow directly from Theorem 1.2 as written. The stress-test note is right about this: Theorem 1.2 is a global-in-time statement, while the corollary is local-in-time, and the necessary scaling/restart argument is absent. Remark 3.3 also says the very weak version is omitted, so the author is aware of the gap.\n\nWhat the paper does well is substantial. Theorem 1.2 gives a clean equivalence between global solvability in L^r(0,∞;X) and a scaling-matched a priori estimate, under Banach-space hypotheses that hold for the concrete Bochner–Besov spaces. The proof is detailed: Lemma 4.1 (the translation-invariance argument) is stated and proved, the Fatou property for Bochner–Besov spaces is proved in Lemma 2.3, and the weak-* compactness is handled explicitly. The applications to Besov spaces (Corollary 1.3) and the by-products on uniqueness and energy equality (Theorem 1.5) are also proved. The dependence on [40] is appropriate—the key lemma is reproduced, not just cited.\n\nThe soft spot is the missing proof of Corollary 4.3, and that is where the abstract's main claim lives. The second part of the corollary (equivalence with the a priori estimate) is also unproved. This is a real gap, not a cosmetic one, because the L^4 case sits exactly at the borderline that Proposition 4.2 cannot reach. But the gap may be repairable: the relaxation idea in Remark 3.3 suggests the author has a route, and the rest of the machinery is in place. I would not desk-reject this.\n\nIf you send it out, ask the referee to focus on the local-to-global reduction and on whether the scaling exponent in Corollary 4.3 is correct. The paper is worth serious referee time: the main theorem is a genuine contribution, and the corollary, once proved, would settle an open question in the generic sense. I'd bring it to reading group while the revision happens.","headline":"Solidly proved main theorem, but the advertised L^4 genericity result depends on Corollary 4.3, which is stated without proof.","tokens_in":29848,"tokens_out":7795,"would_cite":true,"duration_ms":59642,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","76D05","35B65","76D03"],"pacs":[],"model":"deepseek-v4-flash","headline":"Baire-generic L2 data have no L4-integrable weak solutions","keywords":["Navier-Stokes equations","Leray-Hopf solutions","Baire category","scaling invariance","L4 integrability","energy equality","a priori estimates","Besov spaces"],"falsifier":"Construct an open ball in $L^2_\\sigma$ for which every initial datum admits a weak solution with $\\sup_\\nu \\|u^\\nu\\|_{L^4(0,T;L^4)}<\\infty$, or exhibit a uniform-in-viscosity $L^4$ bound with a viscosity exponent different from the one in (10); either observation would contradict Theorem 1.2 and Corollary 4.3.","tokens_in":28852,"feed_emoji":"🌊","tokens_out":10811,"duration_ms":94497,"temperature":0.7,"pith_summary":"This paper asks when weak solutions of the 3D Navier-Stokes equations can land in a prescribed supercritical integrability class $L^r(0,\\infty;X)$. Its central answer is a scaling-driven dichotomy: whenever $5/(3r)+2/s<1$, only a meagre set of $L^2$ divergence-free data admits a weak solution in $\\bigcup_{T>0}L^r(0,T;L^s)$, and in particular a Baire-generic datum has no weak solution in $\\bigcup_{T>0}L^4(0,T;L^4)$. In the complementary window, existence for a non-meagre set is equivalent to a uniform-in-viscosity a priori estimate, so soft existence arguments alone cannot establish such integrability without proving a quantitative scaling bound. If the main theorem is right, global Navier-Stokes regularity would force uniqueness and the energy equality for a residual set of data, and global Euler regularity would force anomalous energy dissipation to fail for a residual set.","feed_headline":"Baire-generic L2 data have no L4-integrable weak solutions","feed_subtitle":"Scaling plus category theory ties such integrability to a uniform viscosity estimate, which generic data fail to satisfy.","key_machinery":"The load-bearing construction is the family of sets $Y_M$ of initial data for which some weak solution satisfies the combined norm bound in (9) and (10). These sets are weakly-star sequentially closed for weak solutions (Proposition 2.17), so if their union is non-meagre, one of them contains a ball. Lemma 4.1, a translation-invariance rescaling lemma, moves that ball to the origin, and the exact $s$-homogeneity of $X$ together with the $2$-homogeneity of $Z$ rescales arbitrary data into it, producing the a priori estimate (10) with the viscosity exponent $3(5/(3r)+2/s-1)$. The same scaling, applied as $\\nu\\to 0$ or to the relaxed Euler system, turns any violation of the critical inequality into a zero solution of the relaxed equations with nonzero data, which is the contradiction that forces the critical-line conditions.","core_discovery":"The paper claims that for suitable $s$-homogeneous Banach spaces $X\\subset S'$ and $2$-homogeneous data spaces $Z\\hookrightarrow L^2_\\sigma$, with $r,s\\in(4/3,\\infty]$ and $1<2/r+3/s<3/2$, integrability in $L^r(0,\\infty;X)$ is controlled by the single scaling exponent $5/(3r)+2/s$. Theorem 1.2 shows that existence of a weak solution in $L^r(0,\\infty;X)$ for a non-meagre set of data is equivalent to the a priori estimate (10), whose viscosity exponent is $-3(5/(3r)+2/s-1)$; this estimate forces $5/(3r)+2/s\\ge 1$, and uniform-in-viscosity bounds force the equality $5/(3r)+2/s=1$. Applying this to $L^r(0,T;L^s)$ gives Corollary 4.3: for a Baire-generic $L^2_\\sigma$ datum no weak solution belongs to $\\bigcup_{T>0}L^4(0,T;L^4)$, and the same mechanism rules out other known sufficient conditions for the energy equality. As a separate application, the Besov classes $L^r(0,\\infty;\\dot B^{\\beta_r}_{r,\\infty})$ with $\\beta_r=(11-3r)/(2r)$ and $5/3\\le r\\le 3$ are shown to be the only scale-compatible classes at the uniform-in-viscosity threshold.","pith_inferences":["Inference: the equivalence between generic existence and a uniform a priori estimate suggests a general meta-principle for scale-invariant PDEs: in supercritical classes, qualitative existence statements secretly contain quantitative bounds, so no-go results of this shape can be converted into necessary estimates that any successful method must prove.","Inference: the critical line $5/(3r)+2/s=1$ is exactly the Euler scaling relation, so the theorem gives a rigorous reason why the energy-dissipating part of an inviscid limit, if it exists, must be looked for at the Onsager threshold rather than in coarser supercritical norms.","Inference: a concrete testable extension would be to monitor the $L^4$ norm in high-resolution Galerkin or spectral truncations from generic data: the theorem predicts that uniform-in-viscosity boundedness can only hold with the specific viscosity exponent of (10), so any observed bound with a different exponent would contradict Corollary 4.3.","Inference: the by-product on residual uniqueness and energy equality suggests that known non-uniqueness constructions for forced or very weak solutions are compatible with a residual set of good behavior in the unforced Leray-Hopf class, provided global regularity holds."],"forward_implications":["For any $r,s$ with $5/(3r)+2/s<1$, only a meagre set of $L^2_\\sigma$ data admits a weak solution in $\\bigcup_{T>0}L^r(0,T;L^s)$; taking $r=s=4$ excludes $L^4(0,T;L^4)$ integrability for Baire-generic data.","Non-meagre global solvability in $L^r(0,\\infty;X)$ is equivalent to the a priori estimate (10), so any proof of such integrability must establish a quantitative viscosity-scaling bound, not merely an existence argument.","Uniform-in-viscosity bounds in any supercritical class are possible only on the critical line $5/(3r)+2/s=1$, which singles out the Onsager-critical Besov spaces $L^r(0,\\infty;\\dot B^{\\beta_r}_{r,\\infty})$ as the natural threshold classes.","If global regularity holds for the Navier-Stokes equations, then for a Baire-generic datum the Leray-Hopf solution is unique and satisfies the energy equality at almost every time; if global regularity holds for the Euler equations, anomalous energy dissipation fails for a Baire-generic datum.","The energy-equality criteria of Lions ($L^4(0,T;L^4)$), Shinbrot, and the Besov-scale criteria are generically unavailable for Leray-Hopf solutions, even though each would be sufficient if it held."],"supporting_citations":[{"why":"Supplies the functional-analytic rescaling lemma used to pass from a non-meagre set of data to a uniform a priori estimate.","marker":"[40]"},{"why":"Gives the original existence theorem for Leray-Hopf solutions for every $L^2$ datum, the object whose integrability is being questioned.","marker":"[47]"},{"why":"Provides the optimal initial-value condition for local strong solutions and the earlier generic exclusion for integrability exponents below $5/4$.","marker":"[32]"},{"why":"Establishes the $L^4(0,T;L^4)$ energy-equality criterion whose generic failure is the flagship corollary.","marker":"[49]"},{"why":"Extends the energy equality to Shinbrot exponents, one of the sufficient conditions ruled out generically.","marker":"[57]"},{"why":"Identifies the Besov class $L^3(0,T;B^{1/3}_{3,\\infty})$ and its energy-conservation threshold, the target of Corollary 1.3.","marker":"[15]"},{"why":"Contains the $H^1$-data a priori estimate used in the proof of Theorem 1.4 and in the global-regularity consequences.","marker":"[61]"}],"fun_headline_variants":["Baire-generic L2 data lack L4 weak solutions","Typical Navier-Stokes data fail L4 integrability","Generic L2 initial data defy L4 integrability","Integrability of NS solutions fails generically in L2","Uniform viscosity bound rules NS integrability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument needs the relevant solution classes to be closed under weak-star limits and the spaces $X$ and $Z$ to be exactly homogeneous under scaling, so that a non-meagre set of good data can be rescaled into a fixed ball; if a natural space is only approximately homogeneous or lacks that closure, the equivalence between generic existence and the a priori estimate may fail.","fun_headline_variants_meta":{"raw":{"variants":["Baire-generic L2 data lack L4 weak solutions","Typical Navier-Stokes data fail L4 integrability","Generic L2 initial data defy L4 integrability","Integrability of NS solutions fails generically in L2","Uniform viscosity bound rules NS integrability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00029,"raw_usage":{"total_tokens":1978,"prompt_tokens":1511,"completion_tokens":467,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":1127,"completion_tokens_details":{"reasoning_tokens":389}},"tokens_in":1127,"tokens_out":467,"duration_ms":5314,"temperature":1.0,"reasoning_tokens":389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:27:58.641254+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an open ball in $L^2_\\sigma$ for which every initial datum admits a weak solution with $\\sup_\\nu \\|u^\\nu\\|_{L^4(0,T;L^4)}<\\infty$, or exhibit a uniform-in-viscosity $L^4$ bound with a viscosity exponent different from the one in (10); either observation would contradict Theorem 1.2 and Corollary 4.3.","supporting_citations":[{"cited_title":"Guerra, L","cited_arxiv_id":null,"evidence_quote":"Supplies the functional-analytic rescaling lemma used to pass from a non-meagre set of data to a uniform a priori estimate."},{"cited_title":"Farwig and H","cited_arxiv_id":null,"evidence_quote":"Provides the optimal initial-value condition for local strong solutions and the earlier generic exclusion for integrability exponents below $5/4$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the $L^4(0,T;L^4)$ energy-equality criterion whose generic failure is the flagship corollary."},{"cited_title":"Cheskidov, P","cited_arxiv_id":null,"evidence_quote":"Identifies the Besov class $L^3(0,T;B^{1/3}_{3,\\infty})$ and its energy-conservation threshold, the target of Corollary 1.3."},{"cited_title":"PDE 6 (2013), no","cited_arxiv_id":null,"evidence_quote":"Contains the $H^1$-data a priori estimate used in the proof of Theorem 1.4 and in the global-regularity consequences."}],"review_version":1}