{"id":"c99d898a-c53b-4718-b7b2-d009411826fe","arxiv_id":"2412.10404","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Infinitely many finite-energy weak solutions to 3D Navier-Stokes in R^3 are built by convex integration with prescribed kinetic energy profiles, extending the torus result of Buckmaster and Vicol.","lead":"This paper constructs infinitely many different finite-energy weak solutions of the 3D Navier-Stokes equations on all of space that start from the same velocity. It extends a famous non-uniqueness result from the periodic box to R^3, using a new localized corrector step in the convex integration method.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exponent balance in the localized-corrector estimates is off by hundreds of powers of λ_q: the forcing F_{q+1} is not small, so Proposition 3.8 cannot hold as written.","rationale":"The reader's verdict was CONDITIONAL, with Proposition 3.8 identified as the weakest assumption. My independent reading confirms that the localized corrector is the critical step, but locates a more specific and checkable failure: the smallness estimates for the forcing are numerically impossible under the stated parameter choices. This is not a matter of outside consensus or missing details; it is an internal exponent inconsistency. If the check confirms it, the central construction collapses and the paper should be rejected unless the parameter definitions or estimates are substantially corrected. I therefore recommend REJECT rather than CONDITIONAL, because the issue affects the core mechanism and is not a cosmetic gap.","tokens_in":50388,"tokens_out":40124,"duration_ms":369614,"concrete_test":"Recompute the exponent balance for the key estimates (3.51) and (3.58) using the definitions in (2.2) and the amplitude bounds in Proposition 3.3. Specifically, substitute ℓ_q = λ_q^{-60} into the right-hand side of (3.51) and (3.58) and evaluate the power of λ_q as q→∞. If the resulting powers are positive (e.g., +1319.97 or +719.97), then ‖F^(1)‖_{B^{-3/2}} cannot be ≤ λ_q^{-40}, and Propositions 3.7 and 3.8 must fail. A full check would also verify the analogous counts in (3.63)–(3.64).","verdict_should_be":"REJECT","load_bearing_attack":"Proposition 3.8 is load-bearing: it produces the localized corrector w^(ns) with ‖w^(ns)‖_{B^{1/2}} ≤ λ_q^{-20}, and the whole iteration depends on the forcing F_{q+1} being small. But the estimates feeding into Proposition 3.7 are arithmetically inconsistent. Using the paper's definition ℓ_q = λ_q^{-60} from (2.2), the right-hand side of (3.51), ℓ_q^{-22} λ_{q+1}^{-1/32}, equals b^{-1/32} λ_q^{1320 - 1/32}, which diverges as q→∞. Likewise, (3.58) gives ℓ_q^{-12} λ_{q+1}^{-1/32} = b^{-1/32} λ_q^{720 - 1/32}, and (3.63)–(3.64) give ℓ_q^{-12} λ_{q+1}^{-9/4} ≈ λ_q^{717.75} and ℓ_q^{-4} λ_{q+1}^4 ≈ λ_q^{244} for F^(2). These quantities are astronomically large, not small. Proposition 3.7 nevertheless concludes ‖F_{q+1}‖_{B^{-3/2}} ≤ λ_q^{-40} and ‖F_{q+1}‖_{H^4} ≤ λ_{q+1}^5. No cancellation mechanism is displayed; these are individual summands with the stated sizes. If this exponent count is correct, the forcing is not small, the fixed-point/continuity argument in Proposition 3.8 has no small data to close, and the compact-support propagation for ˚R_{q+1} breaks. Thus Theorem 1.2 is not established by the present proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a convex integration construction for the 3D Navier-Stokes equations on the whole space R^3, aiming to prove non-uniqueness of weak solutions in C([0,T];L^2(R^3)) with prescribed L^2 energy profiles (Theorem 1.2), and from that, infinitely many dissipative weak solutions (Corollary 1.3). The central novelty is an iterative scheme that splits the approximate solution into a compactly supported local part and a nonlocal part, and introduces a 'localized corrector' w^(ns) defined as the solution of a forced Navier-Stokes-type system (LNS) whose role is to absorb nonlocal, non-divergence errors while preserving compact support of the Reynolds stress. The same scheme is claimed to yield non-uniqueness in bounded domains (Theorem 1.4) and an instability result near Couette flow (Theorem 1.5).","tokens_in":50757,"tokens_out":8571,"duration_ms":81254,"significance":"If the main theorem were established, it would be a major extension of the Buckmaster-Vicol torus construction to the whole space within the finite-energy class, and the local/nonlocal decomposition with the localized corrector is a genuinely interesting idea for handling noncompact spatial domains. The paper is also commendably explicit in its construction of amplitudes, building blocks, and perturbations, and it identifies the specific difficulty of maintaining compact support of the Reynolds stress. However, the proof as written contains a decisive arithmetic error in the size estimates for the forcing term F_{q+1}, which is load-bearing for the entire induction. The claimed results therefore are not established by the present manuscript.","major_comments":[{"comment":"The proof of Proposition 3.7 concludes that ||F_{q+1}||_{B^{-3/2}_{2,1}} is bounded by λ_q^{-40}, but the individual estimates just established do not support this. With ℓ_q = λ_q^{-60} from (2.2) and λ_{q+1}=bλ_q, the term ℓ_q^{-22}λ_{q+1}^{-1/32} in (3.51) equals b^{-1/32}λ_q^{1320 - 1/32}, which diverges as q grows rather than being ≲ λ_q^{-40}. Similarly, the term ℓ_q^{-12}λ_{q+1}^{-1/32} in (3.58) equals b^{-1/32}λ_q^{720-1/32}, and the terms in (3.63)-(3.64) give ℓ_q^{-12}λ_{q+1}^{-9/4} ≈ b^{-9/4}λ_q^{717.75} and ℓ_q^{-4}λ_{q+1}^4 ≈ b^4λ_q^{244}. These are astronomically large, not small, and no cancellation mechanism is displayed. Since Proposition 3.8 relies on the smallness of F_{q+1} to construct the localized corrector w^(ns) with ||w^(ns)||_{B^{1/2}} ≤ λ_q^{-20}, the induction step collapses. This is a load-bearing error in the proof of Theorem 1.2.","section":"Section 3, Proposition 3.7, Eqs. (3.51), (3.58), (3.63)-(3.64)"},{"comment":"The existence and uniqueness statement for the localized corrector w^(ns) is only sketched. The proof says 'By the Banach fixed point theorem and the continuity method' without giving the fixed-point map, the relevant estimates for the nonlinear terms in (LNS), or the precise smallness condition that closes the argument. The argument also uses the smallness of F_{q+1} from Proposition 3.7, which, as noted above, is not established. Even if the forcing were small, the proof of the fixed point would need to be made explicit, because the nonlocal coefficient u^{non-loc}_{ℓq} must be small in B^{1/2}_{2,1} and the quadratic term w^(ns)⊗w^(ns) must be handled with the product law in that Besov scale.","section":"Section 3.2.3, Proposition 3.8"},{"comment":"There is an inconsistency in the stated growth rate. Theorem 1.5 claims ||u_ε(t)-U||_{L^2} ≥ ε^{-1/2} for 3ε^{1/2} ≤ t ≤ 5ε^{1/2}, but the proposition actually proved, Proposition 5.4, only yields ||v_ε(t)||_{L^2} ≥ ε^{-1/4} on the same time interval. Since the proof of Theorem 1.5 is reduced directly to Proposition 5.4, the theorem as stated is not established. This is either a misstatement of Theorem 1.5 or an unproved strengthening in the conclusion.","section":"Section 5.2, Theorem 1.5 and Proposition 5.4"},{"comment":"The proof of Proposition 5.2 (the bounded-domain iteration) is presented as an outline: it states the mollified system, asserts estimates such as (5.11)-(5.13), and then says that 'following the proof as shown in Section 3.3' gives the result. This would be acceptable if the Section 3 estimates were correct, but here the bounded-domain argument explicitly quotes the same decomposition estimates (3.55), (3.63), and the same erroneous ℓ_q^{-22}λ_{q+1}^{-1/32} type bounds, so the same exponent imbalance propagates. A complete proof with corrected estimates is needed before Theorem 1.4 can be considered established.","section":"Section 5.1, Proposition 5.2"}],"minor_comments":[{"comment":"There are several typos: 'Na vier-Stokes' in the title, 'Navier-Stokes equations on torus T3in' with a missing space, and 'victor fields' in the proof of Proposition 3.8. A careful proofreading pass is needed.","section":"Title and abstract"},{"comment":"The sentence 'Readers can refer to [13] for this equality' appears to reference the wrong work; the disjoint-support property of the shifted building blocks is standard but would be better attributed to the correct source or proved in the appendix.","section":"Section 3.2.2, after Eq. (3.37)"},{"comment":"The notation ~L∞_t B^{1/2}_{2,1} is defined in the appendix, but the paper uses it in Section 2 with only a pointer; it would improve readability to define it in the Notations section as well.","section":"Notation, Section 1.4"},{"comment":"The proof of Proposition 3.11 estimates the term I by citing integration by parts with L=60; the displayed bound ℓ_q^{-200}λ_{q+1}^{-1} follows only if the oscillation factor σL is handled consistently. The computation is plausible but should be written out, as this is a key energy-gap estimate.","section":"Section 3.2.5, Proposition 3.11"}],"recommendation":"reject","confidential_remarks":"The manuscript addresses an important question and contains a creative scheme, but the central smallness estimate in Proposition 3.7 is contradicted by the paper's own definitions: powers of ℓ_q^{-1}=λ_q^{60} overwhelm the negative powers of λ_{q+1}, yielding diverging bounds rather than λ_q^{-40}. This is not a minor gap or a constant issue; it breaks the localized corrector step and hence the main induction. The discrepancy between Theorem 1.5 (ε^{-1/2}) and Proposition 5.4 (ε^{-1/4}) further suggests that the applications have not been checked with the same care. I recommend rejection, while noting that the underlying idea might be salvageable with a substantially reworked set of estimates."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper proposes a new mechanism for extending Buckmaster–Vicol convex integration to R^3, but the estimates that make the iteration close are arithmetically wrong. The stress-test note checks out.\n\nWhat is actually new: the split of the approximate solution into local and non-local parts, with the localized corrector w^(ns) defined through a forced Navier–Stokes equation, is a genuine attempt to keep the Reynolds stress compactly supported. That is the right kind of idea for the whole-space problem, and the paper is honest about relying on earlier work for the geometric lemma and building blocks.\n\nThe problem: Proposition 3.7 is load-bearing, and its bounds contradict the preceding estimates. With ℓ_q = λ_q^{-60}, the right side of (3.51), ℓ_q^{-22} λ_{q+1}^{-1/32}, equals b^{-1/32} λ_q^{1320-1/32}, which blows up as q grows. Similarly, (3.58) gives λ_q^{720-1/32} (up to a constant), and (3.63)–(3.64) give λ_q^{717.75} and λ_q^{244}. These are not small; they are huge. Proposition 3.7 nevertheless concludes ‖F_{q+1}‖_{B^{-3/2}} ≲ λ_q^{-40}. No cancellation is shown, and the displayed derivation collects these terms directly. Proposition 3.8's fixed point therefore has no small data to work with. That is not a cosmetic typo; it breaks the compact-support propagation and the whole iteration.\n\nThere are also smaller issues: Theorem 1.5 and Proposition 5.4 disagree on the growth rate (ε^{-1/2} vs ε^{-1/4}), and the bounded-domain proof (Prop 5.2) is only an outline. Those could be fixed if the main argument were sound.\n\nWho should read it: specialists in convex integration who want to see whether the exponent errors can be repaired. The local/nonlocal decomposition is worth a close look, but not as a citable proof.\n\nRecommendation: send it to a serious referee because the problem is important and the approach is inventive; but the referee will find these exponent errors, and the paper as written should not be accepted.","headline":"The local/nonlocal scheme is a real idea, but the core smallness estimates in Prop 3.7 are off by hundreds of powers of λ_q; as written, the main theorem doesn't follow.","tokens_in":51297,"tokens_out":7670,"would_cite":false,"duration_ms":67263,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","76D03"],"pacs":[],"model":"deepseek-v4-flash","headline":"Finite-energy weak solutions of the 3D Navier–Stokes equations on the whole space are non-unique: two solutions can share the same initial data yet follow distinctly prescribed energy profiles.","keywords":["Navier-Stokes equations","non-uniqueness","weak solutions","convex integration","whole space","Reynolds stress","localized corrector","Couette flow instability"],"falsifier":"To test the claim, compute the forcing term $F_{q+1}$ and the Duhamel map in (LNS) for the first iterations with a concrete choice of $e(t)$ and $\\tilde e(t)$, and check whether the Banach fixed point in Proposition 3.8 closes on $[0,T]$ with $\\|w_{q+1}^{(\\mathrm{ns})}\\|_{\\widetilde{L}^\\infty_t B^{1/2}_{2,1}} \\le \\lambda_q^{-20}$; a detectable failure would be any new Reynolds stress $\\mathring{R}_{q+1}$ whose support leaves $\\Omega_{q+1}$, contradicting the induction condition (2.8).","tokens_in":50185,"feed_emoji":"🌊","tokens_out":12653,"duration_ms":102123,"temperature":0.7,"pith_summary":"This paper aims to prove that weak solutions of the three-dimensional incompressible Navier–Stokes equations are not unique in the class of finite-energy weak solutions on the whole space $\\mathbb{R}^3$. The main theorem asserts that for any $T>0$ and any smooth positive energy profiles $e(t)$ and $\\tilde e(t)$ that agree up to time $T/2$, there are weak solutions $u,\\tilde u \\in C([0,T];L^2(\\mathbb{R}^3))$ with equal initial data and $\\|u(t)\\|_{L^2}^2 = e(t)$, $\\|\\tilde u(t)\\|_{L^2}^2 = \\tilde e(t)$. A corollary is the existence of infinitely many energy-dissipating weak solutions. The proof adapts the torus convex integration scheme to $\\mathbb{R}^3$ by splitting each approximate solution into a compactly supported local part and a non-local remainder, with a new localized corrector that keeps the Reynolds stress compactly supported at every step. If the construction is sound, it shows the energy class is far too large for the Cauchy problem of 3D Navier–Stokes to be well-posed.","feed_headline":"3D Navier-Stokes weak solutions are not unique, even in R^3","feed_subtitle":"One initial datum can spawn infinitely many energy-dissipating flows with prescribed kinetic energy profiles.","key_machinery":"The engine of the proof is the iterative convex integration scheme with the decomposition $u_q = u_q^{\\mathrm{loc}} + u_q^{\\mathrm{nonloc}}$, where the local part has compact support inside a shrinking cube $\\Omega_q$ and the nonlocal part lives on all of $\\mathbb{R}^3$. At each step the scheme mollifies $u_q$, then adds a perturbation $w_{q+1}$ built from a principal perturbation using box flows (intermittent building blocks), together with incompressibility, temporal, and temporal-incompressibility correctors, and finally the localized corrector $w_{q+1}^{(\\mathrm{ns})}$, which solves the forced Navier–Stokes problem (LNS) whose forcing $F_{q+1}$ collects the non-divergence and non-compact errors. This corrector is handled by a Banach fixed point argument in the Lerner–Chemin space $\\widetilde{L}^\\infty_t B^{1/2}_{2,1}$, a mixed time-space Besov space, with smallness supplied by the high oscillation frequency of the building blocks. The parameter choices $\\lambda_q = a^{b^q}$, $\\delta_q = \\lambda_q^{-2\\beta}$ with $\\beta = b^{-4}$, make the energy increments summable while each new Reynolds stress error is kept at order $\\delta_{q+1}\\lambda_{q+1}^{-4\\alpha}$ with support in $\\Omega_{q+1}$, so the limit solves the Navier–Stokes equations with the prescribed energy profile.","core_discovery":"The central claim is Theorem 1.2: for any $T>0$ and smooth positive $e(t)$, $\\tilde e(t)$ with $e=\\tilde e$ on $[0,T/2]$, there exist weak solutions $u,\\tilde u \\in C([0,T];L^2(\\mathbb{R}^3))$ of the Navier–Stokes system with $u(0)=\\tilde u(0)$, $\\|u(t)\\|_{L^2}^2=e(t)$, and $\\|\\tilde u(t)\\|_{L^2}^2=\\tilde e(t)$. Choosing profiles that differ after $T/2$ immediately yields distinct weak solutions with identical initial data; choosing monotone decreasing profiles gives infinitely many weak solutions that dissipate kinetic energy (Corollary 1.3). The paper's new contribution is an iterative scheme in which the approximate solution is decomposed as $u_q = u_q^{\\mathrm{loc}} + u_q^{\\mathrm{nonloc}}$, the local part built from intermittent box-flow building blocks and the non-local part corrected by a newly introduced localized corrector $w_{q+1}^{(\\mathrm{ns})}$, defined as the solution of a forced incompressible Navier–Stokes-type equation. This localized corrector absorbs the non-divergence and non-compact errors that the usual inverse-divergence step cannot handle, so the Reynolds stress remains compactly supported and divergence-form at every level. The same scheme then proves non-uniqueness for smooth bounded domains and the $L^2$ instability near the shear flow $(x_2,0,0)$.","pith_inferences":["The local/non-local split should transfer to other non-compact geometries — exterior domains, manifolds with ends, or channels — wherever the Reynolds stress must stay compactly supported while the solution is genuinely nonlocal.","The freedom to prescribe arbitrary smooth energy profiles suggests that no criterion strictly weaker than the Serrin class can restore uniqueness in the energy space, since the construction does not use any special structure of the profiles.","A concrete stress test of the scheme is to run the iteration with the localized corrector omitted and check whether the first new Reynolds stress fails to have support in $\\Omega_{q+1}$; the scheme predicts failure precisely at that check."],"forward_implications":["If Theorem 1.2 is correct, the 3D Navier–Stokes initial-value problem is non-unique in $C([0,T];L^2(\\mathbb{R}^3))$ for every $T>0$: the energy profile alone does not determine the velocity field.","Corollary 1.3 gives infinitely many weak solutions that dissipate kinetic energy, so the standard energy inequality does not enforce uniqueness within the finite-energy class.","Theorem 1.4 extends the same conclusion to smooth bounded domains with Dirichlet boundary conditions, where infinitely many energy-dissipating weak solutions exist.","Theorem 1.5 shows the system is unstable near the shear flow $U=(x_2,0,0)$: for any small $\\epsilon$, a weak solution starts within $\\epsilon$ of $U$ in $L^2$ yet grows to size $\\epsilon^{-1/2}$ on the time window $[3\\epsilon^{1/2},5\\epsilon^{1/2}]$."],"supporting_citations":[{"why":"Supplies the torus convex integration scheme for non-uniqueness of Navier–Stokes weak solutions, which this paper extends to $\\mathbb{R}^3$.","marker":"[14]"},{"why":"Introduces the box flows used to build the principal perturbation and the related building-block estimates.","marker":"[46]"},{"why":"Provides the sharp non-uniqueness framework and the improved Hölder estimate used in Lemma A.4.","marker":"[16]"},{"why":"Supplies the inverse divergence iteration step (Lemma A.5) that keeps the Reynolds stress in divergence form with compact support.","marker":"[12]"},{"why":"Contains the geometric lemma (Lemma A.1) realizing positive definite matrices as sums of $a_k^2\\,\\bar{k}\\otimes\\bar{k}$ and the building-block estimates.","marker":"[6]"},{"why":"Provides the Besov-space and heat-flow smoothing estimates (Lemma A.7) used for the localized corrector.","marker":"[2]"},{"why":"Gives the local well-posedness theory invoked for the bounded-domain version of the iteration.","marker":"[26]"},{"why":"Provides the change of variables and linear estimates used in Proposition A.8 for the Couette flow instability.","marker":"[15]"}],"fun_headline_variants":["Navier-Stokes non-uniqueness extends to R^3","In R^3, weak solutions to Navier-Stokes are non-unique","One start, many energy-dissipating flows in R^3","R^3 Navier-Stokes: infinite weak solutions with same start"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the localized corrector in Proposition 3.8, solving the forced nonlinear problem, exists on the full time interval with the stated smallness bound $\\|w_{q+1}^{(\\mathrm{ns})}\\|_{\\widetilde{L}^\\infty_t B^{1/2}_{2,1}} \\le \\lambda_q^{-20}$; if that fixed point fails, the Reynolds stress cannot be kept compactly supported and the iteration collapses.","fun_headline_variants_meta":{"raw":{"variants":["Navier-Stokes non-uniqueness extends to R^3","In R^3, weak solutions to Navier-Stokes are non-unique","One start, many energy-dissipating flows in R^3","R^3 Navier-Stokes: infinite weak solutions with same start"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00073,"raw_usage":{"total_tokens":3347,"prompt_tokens":1106,"completion_tokens":2241,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":722,"completion_tokens_details":{"reasoning_tokens":2161}},"tokens_in":722,"tokens_out":2241,"duration_ms":17143,"temperature":1.0,"reasoning_tokens":2161,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:03:29.018589+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the claim, compute the forcing term $F_{q+1}$ and the Duhamel map in (LNS) for the first iterations with a concrete choice of $e(t)$ and $\\tilde e(t)$, and check whether the Banach fixed point in Proposition 3.8 closes on $[0,T]$ with $\\|w_{q+1}^{(\\mathrm{ns})}\\|_{\\widetilde{L}^\\infty_t B^{1/2}_{2,1}} \\le \\lambda_q^{-20}$; a detectable failure would be any new Reynolds stress $\\mathring{R}_{q+1}$ whose support leaves $\\Omega_{q+1}$, contradicting the induction condition (2.8).","supporting_citations":[{"cited_title":"B UCKMASTER , V","cited_arxiv_id":null,"evidence_quote":"Supplies the torus convex integration scheme for non-uniqueness of Navier–Stokes weak solutions, which this paper extends to $\\mathbb{R}^3$."},{"cited_title":"M IAO, W","cited_arxiv_id":null,"evidence_quote":"Introduces the box flows used to build the principal perturbation and the related building-block estimates."},{"cited_title":"C HESKIDOV , X","cited_arxiv_id":null,"evidence_quote":"Provides the sharp non-uniqueness framework and the improved Hölder estimate used in Lemma A.4."},{"cited_title":"B EEKIE , T","cited_arxiv_id":null,"evidence_quote":"Contains the geometric lemma (Lemma A.1) realizing positive definite matrices as sums of $a_k^2\\,\\bar{k}\\otimes\\bar{k}$ and the building-block estimates."},{"cited_title":"B AHOURI , J.Y","cited_arxiv_id":null,"evidence_quote":"Provides the Besov-space and heat-flow smoothing estimates (Lemma A.7) used for the localized corrector."},{"cited_title":"F UJITA , T","cited_arxiv_id":null,"evidence_quote":"Gives the local well-posedness theory invoked for the bounded-domain version of the iteration."},{"cited_title":"C HEN , D","cited_arxiv_id":null,"evidence_quote":"Provides the change of variables and linear estimates used in Proposition A.8 for the Couette flow instability."}],"review_version":1}