{"id":"61756172-d2d8-4647-a5f1-6dceeb5d05e6","arxiv_id":"2501.09698","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"This paper constructs nonunique weak solutions of the 3D Navier-Stokes equations in C_t L^q, with 2<q<3, for any prescribed kinetic energy, using L^q-normalized intermittent jets and convex integration.","lead":"The authors prove that the 3D Navier-Stokes equations admit infinitely many weak solutions with a prescribed kinetic energy profile, living in C_t L^q for some q slightly above 2 and starting from zero data. The result extends the convex integration program for Navier-Stokes nonuniqueness to a new regularity class and offers a technical framework aimed at the critical exponent q=3.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The scale-separation condition (3.22) is arithmetically impossible for all admissible parameters, so the iterative estimates collapse.","rationale":"The reader correctly identified parameter feasibility as a weak point, but the decisive issue is more specific: the manuscript itself displays the problematic exponent in (3.22), and that exponent is positive for every admissible parameter range, not merely unexhibited. Since ℓ∼λ_{m+1}^{-5}, the factor ℓ^{-10} is λ_{m+1}^{50} times a constant, so the claimed smallness ℓ^{-10}(λ_{m+1}σ)^{-1/q}≪1 cannot be achieved by any choice of q, ε*, β, b. The same problem propagates into the C^1 estimate in Proposition 3.6 and into Lemma 3.10 through the powers ℓ^{-17}, ℓ^{-35}, ℓ^{-36}. Thus the central induction Proposition 2.1 is not merely unverified; it is contradicted by the displayed algebra. The verdict should be REJECT unless the authors correct the sign/exponent structure and reprove the estimates; a CONDITIONAL verdict waiting only for an explicit tuple would not address this obstruction. This attack is on the argument, not the authors: the text is internally inconsistent as written.","tokens_in":28719,"tokens_out":45435,"duration_ms":437034,"concrete_test":"Recompute ℓ^{-10}(λ_{m+1}σ)^{-1/q} directly from (3.1) and (3.6), without imposing any parameter inequalities. If the λ_{m+1}-exponent is 10[(q-2)(1+ε*)+5+β(b-1)/b]-(3-q-(q+2)ε*)/(3q), then for q∈(2,3), ε*<1/4, β∈(0,1) it exceeds 49.8, so (3.22) is false. Equivalently, instantiate (3.23) with any b satisfying b>60q(5+β(b-1))/(3-q-(q+2)ε*) and β∈(0,1): since τ2<0, the proposed q-upper-bound is below 2, so no admissible q>2 exists.","verdict_should_be":"REJECT","load_bearing_attack":"With ℓ defined in (3.1), using δ_{m+1}^{1/2}/δ_m^{1/2}=b^{-β} and ϑ_{m+1}^{(q-2)/q}=λ_{m+1}^{-(q-2)(1+ε*)}, one obtains ℓ = λ_{m+1}^{-5-(q-2)(1+ε*)} b^{5-β}. Therefore ℓ^{-10}(λ_{m+1}σ)^{-1/q} carries the λ_{m+1}-exponent 10[(q-2)(1+ε*)+5+β(b-1)/b] - (3-q-(q+2)ε*)/(3q), exactly the expression displayed in (3.22). For every q∈(2,3), ε*<1/4, β∈(0,1), this exponent is at least 50 - 1/6 > 0, so the product diverges as m→∞; the required '≪1' cannot hold. The derived parameter condition (3.23) inherits this impossibility: its τ2 = 1 - 120(5+β(b-1)/b) is negative for any b satisfying the accompanying large-b lower bound, making the proposed upper bound on q less than 2. Hence the error estimate in Proposition 3.6, the bound ‖w_{m+1}‖_{C^1} ≤ (1/2)λ_{m+1}^4, Proposition 2.1, and Theorem 1.2 all rest on an impossible smallness estimate. This is not merely a missing explicit parameter tuple: the displayed exponent algebra makes the admissible set empty.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an intermittent convex integration construction of weak solutions to the three-dimensional Navier-Stokes equations in C_t L^q for a uniform exponent 2<q<<3, with arbitrarily prescribed kinetic energy and zero initial data, and without the use of interpolation inequalities. The main iterative statement is Proposition 2.1, which asserts an inductive step for the Navier-Stokes-Reynolds system with the estimates (2.4)-(2.7); Theorem 1.2 is then inferred by passing to the limit. The construction introduces L^q-normalized intermittent jets in Section 3.2, builds a perturbation w_{m+1} in Section 3.3, estimates it in Proposition 3.6, bounds the new Reynolds stress in Lemma 3.10, and closes the energy iteration in Lemma 3.12. The proof follows the standard convex-integration pattern and does not rely on fitted parameters, but the central smallness condition in Proposition 3.6 is arithmetically impossible.","tokens_in":29075,"tokens_out":16917,"duration_ms":168759,"significance":"Conditional significance is high. If Theorem 1.2 were correct, it would provide nonunique C_t L^q weak solutions with q>2, prescribed kinetic energy, and zero initial data, going beyond interpolation-based arguments and moving toward the L^3 critical threshold; the claimed C_t W^{alpha,q} improvement would also be noteworthy. The paper has real strengths: the parameters are governed by explicit inequalities, the prescribed energy is an input rather than a fitted quantity, and the overall scheme follows the established Buckmaster-Colombo-Vicol strategy with L^q-normalized jets. However, the decisive estimate (3.22) has an impossible exponent, and this cannot be repaired by a particular choice of parameters, so the claimed result is not supported.","major_comments":[{"comment":"The decisive smallness condition in (3.22) is arithmetically false. From (3.1), delta_{m+1}^{1/2}/delta_m^{1/2}=b^{-beta}, theta_{m+1}^{(q-2)/q}=lambda_{m+1}^{-(q-2)(1+epsilon*)}, and lambda_m=lambda_{m+1}/b, one obtains ell = lambda_{m+1}^{-5-(q-2)(1+epsilon*)} b^{5-beta}. Hence ell^{-10}(lambda_{m+1}sigma)^{-1/q} = lambda_{m+1}^{10[5+(q-2)(1+epsilon*)] - (3-q-(q+2)epsilon*)/(3q)} b^{-10(5-beta)}. For all admissible q in (2,3) and epsilon*<1/4, the lambda-exponent is at least 50 - 1/6 >0, so the quantity tends to infinity as m grows and can never be much smaller than 1. Even taking the manuscript's displayed exponent in (3.22) at face value, the extra positive term 10 beta(b-1)/b only makes the exponent larger. Therefore the estimate leading to ||w_{m+1}^{(p)}||_{L^q} <= (2/3) kappa_* delta_{m+1}^{1/2} in (3.26) is invalid, and with it Proposition 3.6, Corollary 3.8, Lemma 3.10, Proposition 2.1, and Theorem 1.2 collapse.","section":"Section 3.4, Eq. (3.22) and Proposition 3.6"},{"comment":"The parameter condition (3.23) derived from (3.22) is equally impossible. The desired inequality 10[(q-2)(1+epsilon*)+5+beta(b-1)/b] <= (3-q-(q+2)epsilon*)/(6q) has left-hand side at least 50 while the right-hand side is at most 1/12 for q>2 and 3-q-(q+2)epsilon*<1. No choice of b can satisfy this inequality, and the accompanying lower bound on b does not remove the leading term 5 in the left-hand side. Thus the constraint set used to close the L^q estimate is empty.","section":"Section 3.4, Eq. (3.23)"},{"comment":"Independently of the failure of (3.22), the manuscript does not establish joint feasibility of the parameter constraints (2.7), (3.23), (3.25), (3.28), and (3.34). For example, (3.34) forces b >= 720/(epsilon*-2epsilon) and q <= 2 + (epsilon*-2epsilon)/(4(b+35)(1+epsilon*)), but no explicit tuple is exhibited and no monotonicity argument is given to prove that all constraints can hold simultaneously. Since the inductive step depends on these inequalities to control every term in the Reynolds stress estimate, this is a load-bearing gap in the proof as written.","section":"Section 3.5, Lemma 3.10; Section 2, Proposition 2.1"}],"minor_comments":[{"comment":"The heading 'Proof of Theorem 2.1' appears to be a typo; the proof establishes Proposition 2.1 and then Theorem 1.2.","section":"Section 2, proof of Proposition 2.1"},{"comment":"The notation 'p√11' should be typeset as 11^{1/p} or \\sqrt[p]{11}; as printed it is confusing.","section":"Lemma 3.5"},{"comment":"The function Phi is introduced as a map R^2 -> R^2, but later Phi_sigma and phi_sigma are used as scalar-valued functions in the definitions of the jets; the domain and codomain should be made consistent.","section":"Section 3.2"}],"recommendation":"reject","confidential_remarks":"The main idea is of genuine interest and the paper is written in a standard convex-integration framework, but the exponent error in (3.22) is not a local typo: it invalidates the core perturbation estimate and cannot be fixed by choosing parameters. I recommend rejection in the present form. If the authors can redesign the mollification/oscillation scaling so that ell^{-10}(lambda_{m+1}sigma)^{-1/q} is genuinely small, a fresh submission would be worth considering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The L^q-normalized intermittent jets are a genuinely new ingredient, and the paper is carefully organized around the Buckmaster-Vicol machinery. The bad news: the central scale-separation estimate (3.22) is wrong, and the error is load-bearing. As written, the exponent in (3.22) is always positive for every admissible parameter range, so ℓ^{-10}(λσ)^{-1/q} diverges as m→∞ instead of being ≪1. That estimate is what controls the error term from the improved Hölder inequality in the proof of Proposition 3.6, so the perturbation bound fails and the induction collapses. The reader's worry about a missing parameter tuple undersells the problem: the displayed algebra makes the admissible set empty.\n\nWhat's worth keeping: the L^q-normalized jets and the direct construction in C_t L^q without interpolation are real steps forward in principle, and the energy iteration with prescribed kinetic energy is well thought out. The paper is honest about being inside the Buckmaster-Vicol program, and the parameter discussion shows the authors understand where the threshold q=3 is supposed to enter.\n\nThe soft spots beyond the main error: Remark 1.3 about W^{α,q} is asserted via a reference without a derivation, and the abstract's language about q arbitrarily close to 3 is not supported by the proof, which keeps q near 2. The stress-test note's side claim about τ2 being negative for all admissible b is not needed and may be wrong; the core arithmetic stands on its own.\n\nThis paper is for convex-integration specialists. The idea might be salvageable with a different choice of ℓ or a different jet scaling, but the current proof does not establish Theorem 1.2. I would send it to a serious referee for a careful check of the scaling and parameter constraints; the result is important enough to merit that attention, and the flaw is local rather than conceptual.","headline":"A fresh L^q-normalized jet idea, but a load-bearing arithmetic error at (3.22) makes the main theorem unsupported.","tokens_in":29582,"tokens_out":11734,"would_cite":false,"duration_ms":98409,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","76D05","35D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for a fixed exponent $q$ slightly above $2$, the three-dimensional Navier-Stokes equations admit weak solutions, continuous in time with values in $L^q$, that realize any prescribed smooth nonnegative kinetic energy…","keywords":["Navier-Stokes nonuniqueness","weak solutions","intermittent convex integration","Lq-normalized intermittent jets","prescribed kinetic energy","C_t L^q solutions","zero initial data"],"falsifier":"Check whether the admissible set defined by (2.7), (3.23), (3.25), (3.28) and (3.34) is nonempty: producing a single explicit numerical tuple $(q,\\varepsilon,\\varepsilon_*,A,a,b,\\beta,p)$ that meets all these constraints would validate the consistency of the iteration, while a proof that no such tuple exists would directly falsify Proposition 2.1 and with it Theorem 1.2.","tokens_in":28499,"feed_emoji":"🌊","tokens_out":13216,"duration_ms":113459,"temperature":0.7,"pith_summary":"The paper aims to prove that the three-dimensional Navier-Stokes equations admit far more weak solutions than classical well-posedness theorems suggest: there is a single integrability exponent $q$ slightly above $2$ such that, for any prescribed smooth nonnegative kinetic energy profile $e(t)$, one can find a weak solution $u\\in C_t([0,T];L^q(\\mathbb{T}^3))$ whose squared $L^2$ norm equals $e(t)$ at every time. This matters because it turns the kinetic energy from a quantity that might identify a unique evolution into a freely assignable datum, producing infinitely many distinct solutions all starting from rest. The novelty is that the construction works directly in $L^q$: the building blocks are intermittent jets normalized in $L^q$ rather than $L^2$, so no interpolation inequality is needed, and the solutions inherit a mild fractional spatial regularity $C_t W^{\\alpha,q}$ for some $0<\\alpha\\ll1$. If correct, the result sharply locates where nonuniqueness begins on the integrability scale relative to the critical exponent $q=3$.","feed_headline":"For every smooth energy history, a Navier-Stokes weak solution exists","feed_subtitle":"A convex-integration construction yields time-continuous L^q solutions from rest, with q uniform just above 2.","key_machinery":"The central object is the $L^q$-normalized intermittent jet\n$$\n$W^{{(\\zeta)}}$(x,t)=\\psi_r(N_\\Lambda\\$\\lambda$\\$\\sigma$(x\\cdot\\zeta+\\mu t))\\;\\varphi_\\$\\sigma$\\big(N_\\Lambda\\$\\lambda$\\$\\sigma$(x-\\alpha_\\zeta)\\cdot A_\\zeta,\\,N_\\Lambda\\$\\lambda$\\$\\sigma$(x-\\alpha_\\zeta)\\cdot(\\zeta\\times A_\\zeta)\\big)\\,\\zeta,\n$$\na vector-valued building block with support concentrated on a small set and oscillations at frequency $\\lambda$; it satisfies $\\int|W^{(\\zeta)}|^q\\,dx=1$. The proof uses these jets, rather than $L^2$-normalized Beltrami waves, because they have disjoint supports and carry the $L^q$ normalization explicitly. Their averaged tensor products obey the decomposition identity (3.8), which represents every small symmetric matrix $R$ as a positive combination of $-\\int W^{(\\zeta)}\\otimes W^{(\\zeta)}\\,dx$; this identity is what cancels the previous Reynolds stress in each step. Around this core the iteration adds a time corrector to remove the time derivative of the principal part, an incompressibility corrector to make the velocity divergence-free, and a partition-of-unity energy mechanism that forces the squared $L^2$ norm to follow $e(t)$, including at its zeros.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.2: there exists a uniform exponent $2<q\\ll3$ such that for any nonnegative smooth function $e(t):[0,T]\\to[0,\\infty)$ there is a weak solution $u\\in C_t([0,T];L^q(\\mathbb{T}^3))$ of the periodic Navier-Stokes system with $\\int_{\\mathbb{T}^3}|u(x,t)|^2\\,dx=e(t)$ for every $t$. Taking $e_k(t)=1-\\cos(kt)$ for integers $k\\ge1$ yields infinitely many distinct nontrivial weak solutions all emanating from zero initial data. The proof is constructive: an inductive scheme produces a sequence of smooth solutions of the Navier-Stokes-Reynolds system whose Reynolds stresses decay superexponentially, and the limit solves the true Navier-Stokes equation while tracking the assigned energy. By the fractional Gagliardo-Nirenberg inequalities the constructed solution is further shown to lie in $C_t W^{\\alpha,q}$ for some $0<\\alpha\\ll1$, upgrading the regularity obtained in earlier $L^2$-normalized constructions.","pith_inferences":["It would be worth testing numerically whether an explicit admissible tuple $(q,\\varepsilon,\\varepsilon_*,A,a,b,\\beta,p)$ satisfying all of (2.7), (3.23), (3.25), (3.28) and (3.34) exists; the paper asserts existence but does not display one, and a concrete example would make the iteration computationally checkable.","The same $L^q$-normalized jet construction could plausibly be transplanted to other convex-integration settings, such as transport equations or the Euler equations, where direct $L^q$ control without interpolation might shorten existing arguments.","If the theorem is correct, a consequence the authors do not spell out is an extreme form of nonuniqueness at zero initial data: the kinetic energy alone, at this regularity level, does not select a unique evolution."],"forward_implications":["Every smooth nonnegative kinetic energy profile $e(t)$ is realized by at least one weak solution in $C_tL^q$; in particular $e_k(t)=1-\\cos(kt)$ yields infinitely many distinct nontrivial solutions starting from zero initial data.","The exponent $q$ is fixed once and for all, independent of the chosen profile $e(t)$, so the nonuniqueness is uniform across all energy histories.","The constructed solutions belong to $C_tW^{\\alpha,q}$ for some $0<\\alpha\\ll1$, a spatial-regularity upgrade over earlier $C_tH^\\alpha$ constructions.","The scheme keeps the $L^q$ bound direct, without interpolation inequalities, which the paper proposes as a stepping stone for future convex-integration constructions."],"supporting_citations":[{"why":"[11] supplies the intermittent convex integration scheme and Lemma B.1 (used as Lemma 3.11) for high-frequency error estimates.","marker":"[11]"},{"why":"[7] provides the intermittent jets with disjoint supports that the paper remakes in $L^q$ normalization.","marker":"[7]"},{"why":"[22] introduces the antidivergence operator $\\mathcal{R}$ used to define the Reynolds stress in the convex-integration framework.","marker":"[22]"},{"why":"[20] gives the decomposition lemma for positive definite matrices and the Mikado-flow property of disjoint supports used in Lemma 3.4.","marker":"[20]"},{"why":"[51] establishes the improved Hölder inequality (Lemma 3.7) used to control products of oscillatory building blocks.","marker":"[51]"},{"why":"[6] supplies the fractional Gagliardo-Nirenberg inequalities used to upgrade regularity to $C_tW^{\\alpha,q}$.","marker":"[6]"},{"why":"[23] provides Proposition 4.1 used to estimate derivatives of the coefficient functions in Lemma 3.5.","marker":"[23]"}],"fun_headline_variants":["Prescribed energy yields infinite Navier-Stokes weak solutions","For any given energy curve, a Navier-Stokes weak solution exists","Zero start, infinitely many Navier-Stokes flows","Nonuniqueness: prescribed energy, multiple weak solutions","Convex integration builds solutions for any energy profile"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the joint existence of parameters $(q,\\varepsilon,\\varepsilon_*,A,a,b,\\beta,p)$ satisfying the constraints (2.7), (3.23), (3.25), (3.28), (3.34), together with the auxiliary inequalities in Lemma 3.12; the paper asserts such parameters exist but does not give an explicit tuple, and if the admissible set is empty the inductive construction collapses.","fun_headline_variants_meta":{"raw":{"variants":["Prescribed energy yields infinite Navier-Stokes weak solutions","For any given energy curve, a Navier-Stokes weak solution exists","Zero start, infinitely many Navier-Stokes flows","Nonuniqueness: prescribed energy, multiple weak solutions","Convex integration builds solutions for any energy profile"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000661,"raw_usage":{"total_tokens":3034,"prompt_tokens":972,"completion_tokens":2062,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":1981}},"tokens_in":588,"tokens_out":2062,"duration_ms":16677,"temperature":1.0,"reasoning_tokens":1981,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:45:33.182826+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the admissible set defined by (2.7), (3.23), (3.25), (3.28) and (3.34) is nonempty: producing a single explicit numerical tuple $(q,\\varepsilon,\\varepsilon_*,A,a,b,\\beta,p)$ that meets all these constraints would validate the consistency of the iteration, while a proof that no such tuple exists would directly falsify Proposition 2.1 and with it Theorem 1.2.","supporting_citations":[{"cited_title":"Buckmaster and V","cited_arxiv_id":null,"evidence_quote":"[11] supplies the intermittent convex integration scheme and Lemma B.1 (used as Lemma 3.11) for high-frequency error estimates."},{"cited_title":"Buckmaster, M","cited_arxiv_id":null,"evidence_quote":"[7] provides the intermittent jets with disjoint supports that the paper remakes in $L^q$ normalization."},{"cited_title":"De Lellis and L","cited_arxiv_id":null,"evidence_quote":"[22] introduces the antidivergence operator $\\mathcal{R}$ used to define the Reynolds stress in the convex-integration framework."},{"cited_title":"Daneri and L","cited_arxiv_id":null,"evidence_quote":"[20] gives the decomposition lemma for positive definite matrices and the Mikado-flow property of disjoint supports used in Lemma 3.4."},{"cited_title":"Modena and L","cited_arxiv_id":null,"evidence_quote":"[51] establishes the improved Hölder inequality (Lemma 3.7) used to control products of oscillatory building blocks."},{"cited_title":"Brezis and P","cited_arxiv_id":null,"evidence_quote":"[6] supplies the fractional Gagliardo-Nirenberg inequalities used to upgrade regularity to $C_tW^{\\alpha,q}$."},{"cited_title":"De Lellis and Sz´ ekelyhidi, Dissipative Euler flows a nd Onsager’s conjecture","cited_arxiv_id":null,"evidence_quote":"[23] provides Proposition 4.1 used to estimate derivatives of the coefficient functions in Lemma 3.5."}],"review_version":1}