{"id":"6fcfe146-4a7f-4314-8a25-92ae4eb02b5f","arxiv_id":"2412.16532","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For 3D fractional Navier-Stokes with transport noise and very weak diffusion, α below about 8.7e-9, infinitely many Hölder-continuous Leray-Hopf solutions share one deterministic initial condition up to a positive stopping time.","lead":"This paper constructs infinitely many distinct weak solutions to the 3D fractional Navier-Stokes equations with transport noise, all starting from the same deterministic initial velocity field. It provides the first unforced stochastic setting in which Leray-Hopf solutions are shown to be non-unique.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 8.3's p=∞ endpoint cites Lemma B.8, a positive-Hölder composition estimate, for a negative-regularity Besov bound; the flow-error estimates 6.12/6.15 and the α<α0 range rest on it.","rationale":"The reader's weakest assumption was the bundle of imported flow estimates (4.3)–(4.5) together with the new Besov interpolation Lemma 8.3. My stress-test narrows this to a specific internal proof gap inside Lemma 8.3: the p=∞ endpoint of the interpolation argument invokes a positive-regularity composition lemma (Lemma B.8) for a negative Besov norm. That citation is not valid as written, and no alternative with the same constant is stated. This is load-bearing because Lemma 8.3 controls the difference of the fractional Laplacian under two nearby flows in Proposition 6.12, which in turn feeds Proposition 6.13 and the parameter restrictions of Section 7.4. I did not find a flaw in the convex integration scheme itself: the energy tracking is explicit, the parameter bookkeeping is coherent, the stopping-time construction is standard, and the authors disclose the unusual smallness of α and the local-in-time energy inequality. The manuscript also earns credit for isolating the genuinely new terms (flow error II and mollification error III) and for explaining in Remark 8.1 why simpler Fourier or paralinearization approaches fail. My concern is therefore not that the theorem is false, but that the proof of the key new estimate is, as written, incomplete. A direct re-derivation of Lemma 8.3 would settle the issue: if the claimed L^8 constant and 1/48 exponent survive, the main theorem should stand; if not, the admissible range or the closure of the iteration must be re-examined. Since the reader already conditioned the verdict on verifying exactly these estimates, my read does not change the overall conditional assessment.","tokens_in":100568,"tokens_out":15119,"duration_ms":133066,"concrete_test":"Independently re-derive Lemma 8.3 from first principles for d=3, s=δ−1, ε=1, for both classes of ψ in the statement. In particular, prove the p=∞ endpoint using Lemma B.9 for negative Besov composition instead of Lemma B.8, and track all flow-composition powers of L. If the sharp constant is C_3(L,δ−1,1) ≤ C L^8 with the factor ||ψ−Id||_{L∞}^{1/48}, Propositions 6.12 and 6.15 stand. If the constant or the ψ−Id exponent changes, recompute the stress bounds and recheck the parameter inequalities (7.8), (7.9), (7.20), (7.21), and (7.22): the closure requires ℓ^{−δ−2α}(n+1) ς_n^{γ′/48} to be absorbed by δ_{n+3}^{8β/3} δ_{n+2}, and an enlarged Besov constant would force a smaller α0 or a modified interpolation argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Most of the proof is a careful re-run of [23] with energy tracking, and I found no flaw in the convex integration bookkeeping. The load-bearing point is the new Besov interpolation Lemma 8.3, used in Proposition 6.12 for ˚R_flow^2 and then in Propositions 6.13 and 6.15. In its proof, the p=∞ endpoint is bounded via\n\n||g∘ψ^{-1}-g||_{B^{-s-ε/4}_{∞,∞}} ≤ C (L^{-s-ε/4})^2 ||g||_{B^{-s-ε/4}_{∞,∞}},\n\nwith the citation 'Lemma B.8'. But Lemma B.8 is a composition estimate for positive C^{r+δ} norms; it does not apply to the negative Besov norm B^{-s-ε/4}_{∞,∞}. The paper does not supply a negative-regularity composition bound with the claimed constant. If the correct constant comes from Lemma B.9 after two flow compositions, it is larger in L than the displayed power, and the interpolation constant C_3(L,s,1) may no longer be ≤ C L^8. Because Lemma 8.3 directly controls both the flow error and the mollification error involving (−∆)^α, a larger constant would propagate into the stress estimates and into the parameter choices of Section 7.4, including the admissible interval α < α0 ≈ 8.7·10^{-9}. This is not a criticism of the overall strategy: Remark 8.1 and the authors' own discussion identify this term as the new technical hurdle, and the rest of the manuscript is internally consistent. But the proof of the key lemma, as written, is incomplete at exactly the place the whole construction leans on it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proves that for each sufficiently small α > 0 (specifically α < α0 ≈ 8.7·10^-9), the 3D fractional Navier–Stokes equations on the torus perturbed by Stratonovich transport noise admit infinitely many probabilistically strong, analytically weak Leray–Hopf solutions starting from the same deterministic L2 initial velocity, with paths in C(R+, C^θ) for some θ > α, and distinct on a common strictly positive random time interval. The proof combines a flow transformation that rewrites the SPDE as a PDE with random coefficients, a convex integration scheme adapted from Hofmanová–Lange–Pappalettera [23], and a new Besov interpolation estimate (Lemma 8.3) that controls the error terms arising from the interaction of the fractional Laplacian with the flow. The paper carefully tracks all energy-profile-dependent constants so that the pathwise energy inequality can be closed.","tokens_in":100934,"tokens_out":10999,"duration_ms":87157,"significance":"If correct, this is the first Leray–Hopf non-uniqueness result for the unforced stochastic fractional Navier–Stokes equations, and it also improves the existence theory for analytically weak solutions to (1.1). The proof is exceptionally detailed: the main iterative proposition includes explicit parameter choices, the energy profile dependence is tracked through every constant, and the new Besov interpolation lemma is stated and proved with an explicit constant C_3(L,s,1) ≤ C L^8. The admissible range α < α0 is very small, but the authors are transparent about this limitation. The paper also provides falsifiable predictions in the sense that the constructed solutions have prescribed energy profiles. The main new technical step, Lemma 8.3, is internally consistent; the potential concern about its p=∞ endpoint does not land because the relevant Besov norm has positive regularity.","major_comments":[],"minor_comments":[{"comment":"The p=∞ endpoint of Lemma 8.3 cites Lemma B.8 tersely; since the norm being bounded is B^{-s-ε/4}_{∞,∞} with -s-ε/4 ∈ (0,1), the identification with the classical Hölder space via (1.4) should be stated explicitly so that the positive-regularity application of Lemma B.8 is evident.","section":"Section 8.7.2, proof of Lemma 8.3"},{"comment":"The displayed expression for α0 in (4.16) appears to contain a typo in the denominator; it should match the formula 1/(2bc+1) = [m − 1]/[2(1+ε)(m+ε)^5 + m − 2 − ε] given in Section 7.6.","section":"Equation (4.16) and Section 7.6"},{"comment":"The relations listed after (6.4)–(6.9) are asserted to follow from the definitions, but a short derivation of the most delicate one, (6.8), would improve readability.","section":"Section 6.1, parameter choices"}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about Lemma 8.3 was considered carefully. It appears to stem from misreading the exponent: the Besov norm B^{-s-ε/4}_{∞,∞} has positive regularity because s<0 and −s−ε/2∈(0,1), so Lemma B.8 is applicable via the identification with C^α. The proof is sound at that point, though a brief clarification would help readers. The paper is very long and heavily relies on [23], but that is appropriate for the methods used. Given the significance of the result, I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a real result: the first Leray–Hopf non-uniqueness theorem for the unforced fractional Navier–Stokes equations with transport noise. It removes the deterministic forcing that all earlier stochastic results needed, and it does so via a long, careful convex integration argument that tracks the energy profile through every constant. That is substantial work, and the authors are honest about the limitations—the admissible fractional exponent is about 8.7×10^-9, the energy inequality only holds up to a random stopping time, and the initial datum cannot be prescribed.\n\nThe stress-test flagged Lemma 8.3 as a potential fatal gap. I read that lemma and the surrounding proof carefully, and I think the concern is misplaced. The p=∞ endpoint of the interpolation does not apply Lemma B.8 to a negative-regularity Besov space; because s is negative, −s−ε/4 is positive, and the lemma's composition estimate for positive Hölder norms is exactly what is needed. The displayed power of L is consistent with applying the flow bound twice. So the central load-bearing estimate looks acceptable.\n\nThe soft spots are real but not fatal. The proof relies heavily on imported estimates from [23], and while the authors state them clearly, a referee will need to verify the flow estimates and the constant tracking independently. The parameter choices in Section 7.4 form a delicate web; a small error in the Besov constants would propagate to the admissible range of α, and the current range is already tiny. The result is also local in time for the energy inequality, so the global non-uniqueness question remains open. These are limitations the authors themselves acknowledge, not hidden defects.\n\nWho gets value from this: researchers in stochastic PDEs, convex integration, and the ill-posedness of fluid equations. It is not a paper to assign whole in a reading group, but the main theorem and Lemma 8.3 would be worth discussing. I would not desk-reject this. It deserves a serious referee who checks the imported estimates and the Besov interpolation, even if heavy revisions are expected.","headline":"First Leray–Hopf non-uniqueness for unforced stochastic fractional NSE with transport noise: a serious, detailed proof whose main novelty holds up, though the admissible exponent is minuscule and the energy inequality is local in time.","tokens_in":101467,"tokens_out":4900,"would_cite":true,"duration_ms":42777,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","35Q30","35A02"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that transport noise does not restore uniqueness: for very small fractional diffusion exponents, infinitely many Leray–Hopf solutions can start from the same deterministic velocity field.","keywords":["fractional Navier–Stokes equations","Leray–Hopf solutions","transport noise","non-uniqueness","convex integration","stochastic partial differential equations","flow transformation","Besov spaces"],"falsifier":"Evaluate the norm of the operator $T(h)=h\\circ\\psi^{-1}-h$ in Lemma 8.3 for $d=3$, $s=\\delta+2\\alpha-1$: if the constant grows like $L^{8+\\varepsilon}$ for any $\\varepsilon>0$, the flow-error estimate in Section 8.7.2 cannot be absorbed into $\\delta_{n+2}$ and the construction fails. Equivalently, a direct check of the Wong–Zakai approximation bound (4.3) on $\\mathbb{T}^3$ would test the same load-bearing estimate.","tokens_in":100315,"feed_emoji":"🌊","tokens_out":6884,"duration_ms":57147,"temperature":0.7,"pith_summary":"This paper proves that adding transport noise to the 3D fractional Navier–Stokes equations does not restore uniqueness: for fractional diffusion exponents $\\alpha<\\alpha_0\\approx 8.7\\times 10^{-9}$, it constructs infinitely many Leray–Hopf solutions starting from the same deterministic initial velocity. The solutions are global in time, probabilistically strong, and solve the equation with no additional forcing term. They satisfy the pathwise energy inequality up to a random stopping time, yet any two of them differ on that interval almost surely. This matters because it supplies the first Leray–Hopf non-uniqueness result for the unforced fractional Navier–Stokes equations under any stochastic perturbation, closing a gap left by earlier constructions that needed a specially chosen force.","feed_headline":"Transport noise leaves fractional Navier–Stokes non-unique","feed_subtitle":"Infinitely many Leray–Hopf solutions share one deterministic initial velocity, up to a random time, with no forcing term.","key_machinery":"The proof uses a flow transformation: the stochastic flow $\\Phi$ of the Stratonovich SDE $d\\Phi=\\sum_k\\sigma_k(\\Phi)\\circ dB_k$ conjugates the SPDE (1.1) into a PDE with random coefficients (2.5), involving flowed operators $\\mathrm{div}_\\Phi$, $\\nabla_\\Phi$, and $(-\\Delta)^\\alpha_\\Phi$; solutions are mapped back by $u(t)=v(t)\\circ\\Phi(t)^{-1}$, preserving kinetic energy and regularity. On the transformed equation, a pathwise convex-integration scheme iterates over modified Beltrami waves with an energy-pumping term steering the velocity toward a prescribed energy profile. The genuinely new fractional contributions produce additional flow and mollification errors, controlled by a Besov interpolation lemma (Lemma 8.3) whose dimension-dependent constant satisfies $C_3(L,s,1)\\le C L^8$. This estimate, together with the imported stochastic-flow bounds (4.3)–(4.5), is what forces the extremely small admissible range $\\alpha<\\alpha_0\\approx 8.7\\times 10^{-9}$.","core_discovery":"Theorem 2.4 states that for every $0<\\alpha<\\alpha_0:=1/(2cb+1)$, with $b=38$ and $c$ as in Section 4.2 (numerically $\\alpha_0\\approx 8.7\\times 10^{-9}$), there exists a deterministic initial velocity $u_0\\in L^2(\\mathbb{T}^3)$, an almost surely strictly positive stopping time $\\tau_0$, and infinitely many $\\tau_0$-Leray–Hopf solutions to the stochastic fractional Navier–Stokes system (1.1) with initial condition $u_0$ and paths in $C(\\mathbb{R}_+,C^\\theta(\\mathbb{T}^3))$ for some $\\theta>\\alpha$. Any two of these solutions are distinct on $[0,\\tau_0]$ almost surely. The statement is new because the equation carries no deterministic forcing term, and it constitutes the first Leray–Hopf non-uniqueness result for the unforced fractional Navier–Stokes equations with any stochastic perturbation. The solutions are global in time and satisfy the pathwise energy inequality on $[0,\\tau_0]$.","pith_inferences":["Beyond the paper: if the Besov interpolation constant in Lemma 8.3 can be sharpened, the admissible range of $\\alpha$ should grow well beyond $10^{-8}$, potentially toward the deterministic threshold $\\alpha<1/3$ known in the unforced case.","Beyond the paper: the same flow-transform and convex-integration template is a natural candidate for other dissipative SPDEs with transport noise, provided the analogous flow and mollification errors can be balanced with a milder interpolation loss.","Beyond the paper: a direct numerical check of the Wong–Zakai approximations $\\varphi_n$ against the bound (4.3) on the torus would test the quantitative core of the stochastic-flow step before any further theoretical refinement."],"forward_implications":["For every sufficiently small diffusion exponent $\\alpha$, the class of $\\tau_0$-Leray–Hopf solutions is not a uniqueness class: one deterministic $L^2$ initial condition admits infinitely many distinct solutions.","The pathwise energy inequality holds on the non-empty random interval $[0,\\tau_0]$ for all constructed solutions, so the non-uniqueness persists inside the physically relevant Leray–Hopf subclass.","Restricting to weakly dissipative, transport-noised equations does not eliminate the non-uniqueness phenomenon even when the forcing term used in earlier constructions is removed.","For the wider range $\\alpha<\\tilde\\alpha_0\\approx 1.7\\times 10^{-4}$, the same iteration still produces global solutions with prescribed energy profiles, although without the energy inequality."],"supporting_citations":[{"why":"Supplies the quantitative stochastic-flow estimates (4.3)–(4.5) and the Euler-equation convex-integration scheme that this paper adapts to the fractional setting.","marker":"[23]"},{"why":"Pioneered the convex-integration construction of non-unique Leray–Hopf solutions for hypodissipative Navier–Stokes equations, the pattern followed here.","marker":"[12]"},{"why":"Gives the deterministic fractional-Navier–Stokes Leray–Hopf non-uniqueness template and the energy-profile technique used throughout.","marker":"[15]"},{"why":"Provides the Beltrami-wave building blocks and the geometric lemma used to prescribe the stress and energy in the convex-integration iteration.","marker":"[14]"},{"why":"Proved forced stochastic Leray–Hopf non-uniqueness with multiplicative noise, a baseline that this paper extends to the unforced case.","marker":"[28]"},{"why":"Proved forced additive-noise Leray–Hopf non-uniqueness, another baseline showing that forcing was previously required for such results.","marker":"[7]"}],"fun_headline_variants":["Infinitely many Leray–Hopf solutions from one initial data","Unforced fractional Navier–Stokes with noise: non-unique solutions","Global Leray–Hopf solutions multiply up to random time","First Leray–Hopf non-uniqueness for stochastic fractional N–S"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the imported quantitative stochastic-flow estimates (4.3)–(4.5) and on the Besov interpolation bound $C_3(L,s,1)\\le C L^8$ of Lemma 8.3; if either fails, the flow-error and mollification-error controls in the convex-integration iteration collapse.","fun_headline_variants_meta":{"raw":{"variants":["Infinitely many Leray–Hopf solutions from one initial data","Unforced fractional Navier–Stokes with noise: non-unique solutions","Global Leray–Hopf solutions multiply up to random time","First Leray–Hopf non-uniqueness for stochastic fractional N–S"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1345,"prompt_tokens":939,"completion_tokens":406,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":327}},"tokens_in":555,"tokens_out":406,"duration_ms":3712,"temperature":1.0,"reasoning_tokens":327,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:29:54.284480+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the norm of the operator $T(h)=h\\circ\\psi^{-1}-h$ in Lemma 8.3 for $d=3$, $s=\\delta+2\\alpha-1$: if the constant grows like $L^{8+\\varepsilon}$ for any $\\varepsilon>0$, the flow-error estimate in Section 8.7.2 cannot be absorbed into $\\delta_{n+2}$ and the construction fails. Equivalently, a direct check of the Wong–Zakai approximation bound (4.3) on $\\mathbb{T}^3$ would test the same load-bearing estimate.","supporting_citations":[{"cited_title":"Hofmanov´ a, T","cited_arxiv_id":null,"evidence_quote":"Supplies the quantitative stochastic-flow estimates (4.3)–(4.5) and the Euler-equation convex-integration scheme that this paper adapts to the fractional setting."},{"cited_title":"Colombo, C","cited_arxiv_id":null,"evidence_quote":"Pioneered the convex-integration construction of non-unique Leray–Hopf solutions for hypodissipative Navier–Stokes equations, the pattern followed here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the deterministic fractional-Navier–Stokes Leray–Hopf non-uniqueness template and the energy-profile technique used throughout."},{"cited_title":"De Lellis and L","cited_arxiv_id":null,"evidence_quote":"Provides the Beltrami-wave building blocks and the geometric lemma used to prescribe the stress and energy in the convex-integration iteration."},{"cited_title":"Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations","cited_arxiv_id":"2309.03668","evidence_quote":"Proved forced stochastic Leray–Hopf non-uniqueness with multiplicative noise, a baseline that this paper extends to the unforced case."},{"cited_title":"Non-uniqueness in law of Leray solutions to 3D forced stochastic Navier-Stokes equations","cited_arxiv_id":"2309.09753","evidence_quote":"Proved forced additive-noise Leray–Hopf non-uniqueness, another baseline showing that forcing was previously required for such results."}],"review_version":1}