{"id":"37e37086-631f-454c-8fde-ef6168d4bfd9","arxiv_id":"2505.07181","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"For every divergence-free, mean-free initial condition in L2, the 3D stochastic Navier-Stokes equations with Lipschitz multiplicative noise admit infinitely many global probabilistically strong weak solutions, implying non-uniqueness in law.","lead":"This paper proves that the three-dimensional stochastic Navier-Stokes equations driven by a broad class of multiplicative noises have infinitely many different solutions for any given square-integrable initial state, and also have many different long-term statistical steady states. It matters because it shows that velocity-dependent random noise does not restore uniqueness in 3D fluid equations, a central question in stochastic fluid dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 rests on Proposition 6.5, whose proof in Section 6.1 is explicitly omitted \"for brevity\"; the unverified cut-off and noise estimates are the key new ingredients for the Cauchy problem.","rationale":"The reader's weakest assumption was Assumption 1.1, which is a reasonable hypothesis, but the text already gives parameter-free examples and the proof only uses the Lipschitz and growth bounds; the \"linear operator\" wording in Assumption 1.1 appears to be an error because Example 1.2.3 is nonlinear, yet Proposition 1.9 is claimed to satisfy the assumption. This is a local inconsistency, not the core risk. The core risk is that the proof of the Cauchy-problem iteration, Proposition 6.5, is explicitly omitted and contains exactly the new technical features needed for Theorem 1.2. Without a complete proof, the central claim is conditional. Since the overall structure is plausible and consistent with prior work, the appropriate verdict remains CONDITIONAL; no change from the reader's assessment.","tokens_in":44149,"tokens_out":22496,"duration_ms":199812,"concrete_test":"Independently complete the proof of Proposition 6.5 by deriving the bound on ˚R_cut in Section 6.1.3: compute ∥(1/µ)(χ²_{q+1})′ Σ PP≠0 a² φ² ψ² ξ∥_{C_t L1} and ∥χ′_{q+1}(w^(p)+w^(c))∥_{C_t L1} on [κ_{q+1}, 2κ_{q+1}] using (A.5) and (5.14), and verify that the claimed ℓ^{-20} λ^{-1/8+3ε} ≤ ℓ^γ and the matching powers are consistent with (6.5)-(6.6). Then check that Lemma 6.11 indeed yields |||˚R_{q+1}|||_{L1,r0,2κ_{q+1}} ≤ c_R ε_q and that κ_q Λ_{q-1}^{r0} ≤ ε_{q-1} follows from a valid choice of N0. If all these exponent balances close, the omitted-proof concern is resolved; if not, Theorem 1.2 is unverified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2 depends on Proposition 6.5, the Cauchy-problem iteration step, but Section 6.1 states that \"the remaining constructions and estimates follow similarly to the previous case and we kindly omitted for brevity.\" This is an explicit omitted proof at the exact point where the initial condition is handled. The new cut-off function χ_q, the deterministic initial part ˚z, the time scale κ_q = ℓ_q^{1/4}, and the modified parameter relations (6.5)-(6.6) introduce stress terms ˚R_cut and noise estimates that are used in Lemmas 6.7, 6.9, 6.11, and 6.12 but are not actually derived. In particular, the bound (6.23) for ˚R_cut and its use to obtain (6.24) in Lemma 6.11 require checking exponent balances, for example ℓ^{-20} λ^{-1/8+3ε} ≤ ℓ^γ under (6.6); and the proof of Theorem 1.2 needs κ_q Λ_{q-1}^{r0} ≤ ε_{q-1}, which depends on a choice of N0 not exhibited. If any of these steps fails, the constructed v_q may not satisfy Assumption 6.1 and the limit u = v + z may not solve (1.1) with initial datum u0. This is a completeness gap, not an objection to the underlying method.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the 3D incompressible Navier–Stokes equations on the torus driven by multiplicative noise, under Assumption 1.1, which requires the noise coefficient G to be Lipschitz from L^{p0}, p0∈[1,2), into Hilbert–Schmidt operators with values in H^{-2δ0}, δ0∈[0,1/2). The main results are: Theorem 1.2, existence of infinitely many probabilistically strong, analytically weak global solutions for every divergence-free mean-free L^2 initial condition, implying non-uniqueness in law (Corollary 1.3); Theorem 1.6, existence of solutions with prescribed energy profile; and Theorem 1.7, existence of infinitely many ergodic stationary solutions. The proofs are based on a stochastic convex integration scheme: the solution is split into a stochastic convolution z and a deterministic-like part v, with an iteration in which the noise is re-solved at each step and the Reynolds stress is controlled by stochastic estimates. Section 5 gives a detailed proof of the iteration for the stationary-solution construction, while Section 6 treats the Cauchy problem by adding a time cut-off and a deterministic initial part; the proof of Proposition 6.5, the main iteration for the Cauchy problem, is substantially abbreviated. I assess that the core strategy is sound and the Section 5 estimates are presented in detail, but the manuscript as written does not contain a complete proof of Theorem 1.2.","tokens_in":44475,"tokens_out":17146,"duration_ms":161147,"significance":"If the proof is completed, the paper would be a substantial advance in the stochastic convex integration literature. Prior results for multiplicative noise required much more regular coefficients; here the noise coefficient only needs a Lipschitz/linear-growth bound in L^{p0} with values in a rough Sobolev space. The stochastic convolution estimates in Section 3 and the iteration estimates in Section 5 are detailed, and the construction of ergodic stationary solutions via prescribed energy profiles and Krylov–Bogoliubov is a natural and valuable contribution. The manuscript is not merely an application of existing deterministic convex integration: the main new point is the treatment of the Itô noise contributions through moment estimates rather than pathwise estimates, and Section 5 provides a credible implementation of that idea. However, the main theorem for the Cauchy problem rests on an incompletely proved iteration, so the significance can only be fully credited after the missing arguments are supplied.","major_comments":[{"comment":"Assumption 1.1 states that G is a linear operator from L^{p0} to L2_0(U;H^{-2δ0}), but the example in §1.2.3 and Proposition 1.9 allow nonlinear Nemytskii coefficients g_k satisfying only Lipschitz and linear growth. This is a contradiction in the statement of the main assumption. The proofs in Sections 3–6 use only the Lipschitz bound (1.2) and linear growth (1.3), never linearity itself, so the intended assumption is evidently that G is a general Lipschitz mapping. The assumption should be reworded accordingly, otherwise Theorem 1.2 does not cover the nonlinear examples advertised in the paper.","section":"§1.1, Assumption 1.1; §1.2.3, Proposition 1.9"},{"comment":"The proof of Proposition 6.5, which is the load-bearing iteration for the Cauchy problem and hence for Theorem 1.2, is not supplied. The text at the start of §6.1 states that 'the remaining constructions and estimates follow similarly to the previous case and we kindly omitted for brevity.' This is not an acceptable omission here: the cut-off function χ_q, the deterministic initial part ˚z, the time scale κ_q=ℓ_q^{1/4}, and the modified parameter relations (6.5)–(6.6) introduce genuinely new stress terms such as ˚R_cut, and Lemmas 6.7, 6.9, 6.11 and 6.12 depend on estimates that are asserted rather than derived. In particular the bound (6.23) for ˚R_cut and its use to obtain (6.24) are the exact points where the initial datum is handled; a complete derivation of these estimates must be included.","section":"§6.1, Proposition 6.5"},{"comment":"The parameter inequalities used to control the cut-off stress are not established, and one displayed inequality is impossible as written. The text chooses ε>0 such that λ^{3ε}_{q+1}<ℓ^α_{q+1}; since λ_{q+1} is large and ℓ_{q+1} is small, this inequality cannot hold for any positive ε. The surrounding bound ℓ^{-20}λ^{-1/8+3ε}≤ℓ^γ needs a concrete verification using the definition of λ_q in Definition 6.1, and the compatibility of (6.6) with (4.3) for all admissible p0,δ0 is not evident. Because these inequalities are used in Lemma 6.9 and Lemma 6.11, the parameter regime should be stated explicitly, for example by using the fact that Assumption 1.1 with exponent p0 implies the same assumption with any larger p0<2.","section":"§6.1.3, Eq. (6.23) and surrounding text"},{"comment":"The passage to the limit in the proof of Theorem 1.2 requires the bound κ_q Λ_{q-1}^{r0} ≤ ε_{q-1} in the estimate for ∫_0^T E∥R_q(t)∥^{r0}_{L1}dt. This is not proved, and the choice of N0 (or M0) that would make it true is not exhibited. The base case Proposition 6.4 is also stated without proof, although it must verify Assumption 6.1 for q=1, including the energy condition (6.10), which involves θ_0 and ε_{-1} that are not defined in Definition 6.1. These are not merely cosmetic issues: without an explicit choice of N0 and a verified base case, the convergence of the iteration and the existence of the limit solution in Theorem 1.2 are incomplete.","section":"§6.2, proof of Theorem 1.2"}],"minor_comments":[{"comment":"In the sentence 'see also [BFHM19, ?] and [HZZ23b]', the reference '[BFHM19, ?]' contains a literal question mark; this should be corrected to a proper citation.","section":"§1, Introduction"},{"comment":"The internal numbering of Assumption 6.1 is inconsistent with the lemmas: Lemma 6.10 refers to Assumption 6.1(5), and Lemma 6.12 refers to Assumption 6.1(4), while Assumption 6.1 has four numbered items and the energy condition is item (3). The references should be corrected.","section":"§6.1.3 and §6.1.4"},{"comment":"There are numerous typographical errors, including 'Naiver-Stokes' for 'Navier-Stokes', 'nontation' for 'notation' in §2.1, and inconsistent hyphenation of 'Hölder'. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"In the proof of Theorem 1.7, the line 'For R_N := M N^2, M,N∈N' should probably read 'R_N := M N^2 with a fixed parameter M that is later chosen large'; as written M is not quantified before its use in the display below.","section":"§4.3, tightness estimate"},{"comment":"The cut-off function χ_q is defined for all real t, but the paper only constructs solutions on [0,∞); the definition should state that χ_q is extended by zero for t<0 consistently with the convention z_q(t)=0 for t<0.","section":"§6.1, Definition 6.8"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the incomplete proof of Proposition 6.5 in Section 6.1; the stress-test concern raised about exactly this point is valid. I do not see a fundamental flaw in the overall method—Section 5 is detailed and plausible, and the missing pieces in Section 6 appear to be fillable—but the manuscript should not be accepted with the main Cauchy-problem iteration asserted rather than proved. The inconsistency in Assumption 1.1 between 'linear operator' and the nonlinear examples is also a statement-level problem that must be fixed. I would recommend a major revision, not a rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The main theorem—non-uniqueness in law for 3D stochastic Navier–Stokes with noise coefficients that are Lipschitz from L^{p0} into H^{-2δ0}—is a genuine step beyond [HZZ24], not a repackaging. And the proof has a real completeness gap: the Cauchy problem iteration (Section 6.1) is explicitly abbreviated, and a couple of exponent checks that the argument relies on are not shown. If I had to bet, the gap closes; but the preprint as written does not let you verify the main theorem from beginning to end.\n\nWhat I like. Assumption 1.1 is broad enough to include linear multiplicative, matrix-valued, and Lipschitz Nemytskii noise, and the examples are clearly worked out. Section 3's stochastic convolution bounds are careful and the parameter choices in Section 5 are explicit, with most of the iteration actually proven. The stationary solution part (Theorem 1.7) is standard Krylov–Bogoliubov once you have Theorem 1.6, and it's executed neatly. No circularity: the construction is self-contained.\n\nWhere it gets soft. Section 6.1 says the remaining constructions and estimates 'follow similarly' and are omitted for brevity. That's the exact point where the initial condition forces a new cut-off χ_q, a deterministic part ˚z, a different time scale κ_q = ℓ_q^{1/4}, and modified parameter relations (6.5)-(6.6). Lemmas 6.7–6.12 are stated with proofs, but a lot of 'similarly' hides the actual estimates for the cut-off stress ˚R_cut. The specific step that worries me: in the proof of Theorem 1.2, the bound E∫_0^T ∥R_q∥^{r0} ≲ ε_{q-1} uses κ_q Λ_{q-1}^{r0} ≲ ε_{q-1}. That does not follow from the displayed relations without a choice of N0 that is never exhibited. Likewise (6.23) for ˚R_cut and its use to get (6.24) need an exponent balance ℓ^{-20} λ^{-1/8+3ε} ≤ ℓ^γ under (6.6); plausible, but not shown. These are fillable, but they are load-bearing in the proof of Theorem 1.2. Minor stuff: a few typos and a broken citation '[BFHM19, ?]' in Section 1.\n\nWho this is for. Specialists in stochastic PDEs and convex integration. It deserves a serious referee: the result is important and the method is credible. The referee should ask the authors to provide the omitted Section 6.1 details before publication.","headline":"A genuine extension of stochastic convex integration to general multiplicative noise, with a real but likely fillable gap in the Cauchy-problem iteration (Section 6.1).","tokens_in":45011,"tokens_out":4534,"would_cite":true,"duration_ms":39809,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","35R60","35Q30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under a Lipschitz condition on the noise, the stochastic 3D Navier–Stokes equations admit infinitely many global probabilistically strong, analytically weak solutions, so uniqueness in law fails.","keywords":["stochastic Navier–Stokes equations","non-uniqueness in law","probabilistically strong solutions","convex integration","multiplicative noise","ergodic stationary solutions","intermittent jets","Itô calculus"],"falsifier":"Take a multiplicative noise $G(u) = \\sum_k \\sigma_k \\langle u, e_k\\rangle e_k$ with coefficients chosen so that $G$ maps into $H^{-s}$ only for some $s \\ge 1/2$, while the series $\\sum_k \\sigma_k^2 |k|^{4\\delta_0}$ fails to converge for any $\\delta_0<1/2$. Then compute the stochastic convolution $z_c$ of equation (3.1) and check the bound $\\sup_{t\\ge 0} E\\|z_c\\|^r_{C^\\gamma_t H^{2\\delta}} < \\infty$ claimed in Lemma 3.1. The kernel $(t-s)^{-2(\\delta+\\delta_0)}$ becomes non-integrable at $s=t$, so the bound fails for every choice of the dissipative constant $c$, and the construction has no starting point.","tokens_in":43890,"feed_emoji":"🌊","tokens_out":4871,"duration_ms":47740,"temperature":0.7,"pith_summary":"This paper studies the three-dimensional Navier–Stokes equations driven by a general multiplicative noise. It claims that for any divergence-free, mean-zero initial condition in $L^2$, there exist infinitely many global-in-time solutions that are weak in the analytic sense but strong in the probabilistic sense, so uniqueness in law fails. It also claims the existence of infinitely many ergodic stationary solutions. The result matters because previous non-uniqueness theorems required special noise structures, such as additive, transport, or very regular multiplicative noise, whereas this work replaces those restrictions by a broad Lipschitz condition with a roughness allowance. If the claims are correct, adding general multiplicative noise does not restore uniqueness in law for the 3D Navier–Stokes equations.","feed_headline":"Noisy 3D Navier–Stokes has infinitely many strong solutions","feed_subtitle":"Even with a general multiplicative noise, the probability law fails to pick a unique flow: infinitely many solutions share the same…","key_machinery":"The method is stochastic convex integration. At each iteration step $q$, the approximate velocity $v_q$ and the Reynolds stress $\\mathring{R}_q$ solve a random PDE (4.1), where the stochastic part is a truncated stochastic convolution $\\bar z_q = \\Pi_{\\zeta_q} z_q$ obtained by solving the Itô SPDE (4.2). The iteration adds a perturbation $w_{q+1}$ built from intermittent jets with frequencies $\\lambda_{q+1}$, $r_\\perp$, $r_\\parallel$, $\\mu$, and the amplitude is chosen so that the principal part $w^{(p)}_{q+1}$ cancels the Reynolds stress through identity (5.12). Moment estimates for the stochastic convolution (Lemma 3.1, parameter condition (4.3)) replace the pathwise bounds used in deterministic convex integration, and the energy profile is pinned indirectly through the quantity $\\theta_q$ in (4.9). The Reynolds stress is driven to zero geometrically, and the limit $u = v + z$ is the desired solution.","core_discovery":"The central discovery is Theorem 1.2: under Assumption 1.1, for any divergence-free, mean-zero $L^2$ initial condition independent of the driving Wiener process, there exist infinitely many analytically weak and probabilistically strong solutions to (1.1) in the sense of Definition 1.1. Consequently, non-uniqueness in law holds for every initial law supported on divergence-free, mean-free $L^2$ fields (Corollary 1.3). The construction also yields, for any sufficiently large constant $K$, an ergodic stationary solution with $\tilde{E}\\|\\tilde{u}\\|^2_{L^2}=K$, and infinitely many such solutions exist by varying $K$ (Theorem 1.7).","pith_inferences":["The roughness threshold $\\delta_0 < 1/2$ in Assumption 1.1 is plausibly sharp: if the noise maps into $H^{-s}$ with $s \\ge 1/2$, the key stochastic-convolution bound in Lemma 3.1 cannot hold because the parameter condition $0<\\gamma+\\delta+\\delta_0<1/2-2/r$ becomes empty, suggesting non-uniqueness may stop at $H^{-1}$ noise.","The stochastic convex-integration scheme is modular in the sense that the same iteration could be adapted to other dissipative SPDEs, such as the 2D Navier–Stokes equations with derivative noise or stochastic MHD, provided an analogue of Lemma 3.1 is proved.","The energy-pinning mechanism suggests a stronger phenomenon than non-uniqueness in law alone: the constructed solutions are 'wild' in the deterministic convex-integration sense, with prescribed energy profiles, so the set of solutions is large enough to support ergodic measures with arbitrary mean energy.","A natural testable question is whether the transition from uniqueness to non-uniqueness in law occurs exactly at the critical noise regularity $\\delta_0 = 1/2$, analogous to the deterministic $L^2$-critical threshold for the Navier–Stokes equations."],"forward_implications":["For every divergence-free, mean-zero $L^2$ initial condition, there are infinitely many global probabilistically strong and analytically weak solutions.","Non-uniqueness in law holds for arbitrary initial laws supported on divergence-free, mean-free $L^2$ fields.","For any sufficiently large constant $K$, there exists an ergodic stationary solution with prescribed mean-square energy $K$, and infinitely many such stationary solutions exist.","Given a smooth energy profile $e(t)$, one can prescribe the full evolution $E\\|u(t)\\|^2_{L^2} = e(t)$ for one of the constructed solutions."],"supporting_citations":[{"why":"Provides the deterministic convex-integration method for non-uniqueness of weak Navier–Stokes solutions, which this paper adapts to the stochastic setting.","marker":"[BV19b]"},{"why":"Establishes non-uniqueness in law for stochastic 3D Navier–Stokes with additive or sufficiently regular multiplicative noise, the result that the present paper generalises.","marker":"[HZZ24]"},{"why":"Introduces the notion of probabilistically strong and analytically weak solutions with non-unique Markov selections, the type of solution constructed here.","marker":"[HZZ23a]"},{"why":"Develops stochastic convex integration for sharp non-uniqueness with additive noise, supplying the stochastic convex-integration technique that this paper extends to general multiplicative noise.","marker":"[CDZ22]"},{"why":"Supplies the intermittent jets, the inverse divergence operator, and the purely deterministic building blocks used in the perturbation step.","marker":"[BV19a]"},{"why":"Provides sharp non-uniqueness estimates for deterministic Navier–Stokes, including the bilinear operator $B$ and inverse divergence bounds used in the Reynolds-stress estimates.","marker":"[CL22]"},{"why":"Gives the non-unique ergodicity and stationary-solution construction strategy, including the energy-pinning iteration that this paper reuses in the stochastic setting.","marker":"[HZZ22]"}],"fun_headline_variants":["Stochastic 3D Navier–Stokes has infinitely many solutions in law","Noise can't make 3D Navier–Stokes unique: infinite solutions","Infinite strong solutions for 3D Navier–Stokes under general noise","3D Navier–Stokes with noise: non-uniqueness in law proven","Even with noise, 3D Navier–Stokes admits infinitely many solutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on Assumption 1.1: the noise coefficient must be a Lipschitz linear operator mapping $L^{p_0}$ with $p_0\\in[1,2)$ into Hilbert–Schmidt operators with values in the rough space $H^{-2\\delta_0}$ with $\\delta_0<1/2$, satisfying linear growth. If the noise maps into rougher Sobolev spaces, or the Lipschitz constant is not uniform, the key stochastic-convolution estimates of Lemma 3.1 collapse and the iteration cannot be controlled.","fun_headline_variants_meta":{"raw":{"variants":["Stochastic 3D Navier–Stokes has infinitely many solutions in law","Noise can't make 3D Navier–Stokes unique: infinite solutions","Infinite strong solutions for 3D Navier–Stokes under general noise","3D Navier–Stokes with noise: non-uniqueness in law proven","Even with noise, 3D Navier–Stokes admits infinitely many solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000286,"raw_usage":{"total_tokens":1594,"prompt_tokens":768,"completion_tokens":826,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":384,"completion_tokens_details":{"reasoning_tokens":718}},"tokens_in":384,"tokens_out":826,"duration_ms":7449,"temperature":1.0,"reasoning_tokens":718,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:23:10.778114+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a multiplicative noise $G(u) = \\sum_k \\sigma_k \\langle u, e_k\\rangle e_k$ with coefficients chosen so that $G$ maps into $H^{-s}$ only for some $s \\ge 1/2$, while the series $\\sum_k \\sigma_k^2 |k|^{4\\delta_0}$ fails to converge for any $\\delta_0<1/2$. Then compute the stochastic convolution $z_c$ of equation (3.1) and check the bound $\\sup_{t\\ge 0} E\\|z_c\\|^r_{C^\\gamma_t H^{2\\delta}} < \\infty$ claimed in Lemma 3.1. The kernel $(t-s)^{-2(\\delta+\\delta_0)}$ becomes non-integrable at $s=t$, so the bound fails for every choice of the dissipative constant $c$, and the construction has no starting point.","supporting_citations":[],"review_version":1}