{"id":"ae61cd04-3e28-428b-96b5-aeea47fb2313","arxiv_id":"2504.16687","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Weak solutions of the Euler and Navier-Stokes equations exist for which two different particle trajectories start from the same point, in sharp regularity ranges, both with and without Brownian noise.","lead":"This paper constructs weak solutions of the Euler and Navier-Stokes equations whose fluid particle paths are not unique, both for deterministic trajectories and for trajectories with Brownian noise. It pinpoints sharp regularity boundaries at the Onsager 1/3 exponent and in the supercritical L^p range for Navier-Stokes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Trevisan superposition-principle step is plausibly within its hypotheses, and the β≤tildeβ reduction in Theorem 1.2(2) is justifiable by monotonicity.","rationale":"The reader's weakest assumption focuses on the Trevisan superposition principle, and this is exactly the step that should be checked against the cited source. However, the manuscript already provides more integrability than the standard L^1(μ) hypothesis and the required weak continuity; the missing piece is only the precise statement of [Tre14, Section 7.2]. The reader's secondary concern about β≤tildeβ in Theorem 1.2(2) can be resolved by monotonicity because the strict inequality leaves room to increase tildeβ to at least β. The central convex-integration construction in Sections 3 and 5 is elaborate and mostly internally consistent; the few terse 'without loss of generality' steps are justifiable, and the one interpolation overstatement in the proof of Theorem 1.7 is not load-bearing. Therefore a conditional reading is reasonable but no change of verdict is needed.","tokens_in":91355,"tokens_out":23913,"duration_ms":241711,"concrete_test":"Check the precise hypotheses of Trevisan [Tre14, Section 7.2] for the Fokker-Planck equation: verify that drift integrability ∫∫|v|ρdtdx<∞, which follows from v,ρ∈L^2_{t,x}, is sufficient, and determine whether the extra |v|^{1+ε}ρ condition is required. If L^1(dt;dμ_t) drift integrability is sufficient, Theorem 1.7 follows as stated and the interpolation claim about L^{2+2ε} can be dropped.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern found. The two most plausible weak points both appear repairable. First, the application of Trevisan's superposition principle in the proof of Theorem 1.7: the paper checks the stronger condition ∫∫|v|^{1+ε}ρ<∞ and weak continuity of t↦ρ(t)dx. The weaker condition ∫∫|v|ρ<∞, which is the standard drift-integrability hypothesis for such a superposition principle, follows immediately from v,ρ∈L^2([0,T]×T^d) by Cauchy-Schwarz. Thus, unless [Tre14, Section 7.2] requires an additional condition not stated in the manuscript, the step is valid. Second, the 'without loss of generality' assumption β≤tildeβ in Section 2 is terse but valid: since 0<β<1/3 and β+2tildeβ<1, one can choose tildeβ0 with β<tildeβ0<(1-β)/2, run the construction with (β,tildeβ0), and obtain ρ∈C^tildeβ0⊂C^tildeβ. A minor blemish is the unproved claim in Theorem 1.7 that v∈L^{2+2ε}([0,T]×T^d) follows by interpolation from (4.4)-(4.7); this is not clear for p<2, but it is unnecessary because the required integrability for the superposition principle follows from the L^2_{t,x} bounds alone.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops two convex-integration constructions linking Eulerian non-uniqueness of scalar equations to non-uniqueness of Lagrangian trajectories for the Euler and Navier-Stokes equations. In the Hölder scale, Theorem 1.2 constructs, for β < 1/3 and β + 2β̃ < 1, a C^β weak solution v of the 3D Euler equations with prescribed strictly positive kinetic energy e(t), together with a non-constant positive density ρ ∈ C^{β̃} solving the transport equation with initial data ρ0 = 1; since ρ ≡ 1 is always a solution, this gives two positive solutions, and via the superposition principle it yields non-unique deterministic Lagrangian trajectories on a positive-measure set (Corollary 1.3). In the Sobolev/Lebesgue scale, Theorems 1.10 and 1.7 construct, for d ≥ 2 and (p,r,s) in the set A, a divergence-free solution v ∈ L^r_t L^p ∩ L^2_{t,x} ∩ L^1_t W^{1,s} ∩ C_t L^1 of the Euler or Navier-Stokes equations together with a non-constant positive solution ρ to the Fokker-Planck equation, and then use Trevisan's superposition principle to prove non-uniqueness in law of stochastic Lagrangian trajectories for a positive-measure set of initial points. The sharpness statements are anchored to the Krylov-Röckner uniqueness result for C_t L^p, p > 2, and the Caravenna-Crippa uniqueness result for W^{1,s}, s > d.","tokens_in":91664,"tokens_out":38729,"duration_ms":341698,"significance":"If correct, the results are a substantial contribution to the ill-posedness theory of hydrodynamic equations below the LPS/Onsager regularity thresholds: they upgrade Eulerian non-uniqueness to Lagrangian non-uniqueness, cover the sharp supercritical range up to known endpoints, and give new statements for stochastic trajectories with hydrodynamic drift. The paper is carefully structured, with explicit parameter choices in Sections 3 and 5 and complete appendices for the gluing, Mikado, and intermittency estimates. The construction is a direct existence proof, and the sharpness claims use independent uniqueness theorems rather than circular reasoning. The two most delicate external ingredients, the reduction to β ≤ β̃ in Proposition 2.1 and the Trevisan superposition-principle step in Theorem 1.7, are terse but repairable; on reading the manuscript I do not find them to be load-bearing errors.","major_comments":[],"minor_comments":[{"comment":"The theorem states the conclusion with Hölder exponents C^β and C^{β̃}, but the iteration only yields v ∈ C^{β'} and ρ ∈ C^{β''} for every β' < β and β'' < β̃. Since the hypotheses β < 1/3 and β + 2β̃ < 1 are open, the stated version follows by a standard exponent-boosting argument; adding one sentence explaining this would remove an apparent gap.","section":"Section 2, Proof of Theorem 1.2"},{"comment":"The reduction 'without loss of generality, additionally assume β ≤ β̃' should be justified explicitly: when β̃ < β, choose β̃0 with β < β̃0 < (1−β)/2, apply the construction with (β,β̃0), and note that C^{β̃0} ⊂ C^{β̃}; the theorem is not symmetric, so this step is not automatic.","section":"Section 2, Proposition 2.1 and its hypotheses"},{"comment":"The application of [Tre14, Section 7.2] should be made self-contained: state the precise hypotheses of the superposition principle being used and verify them. In particular, the first-moment condition ∫∫ |v(s,x)| ρ(s,x) dx ds < ∞ follows immediately from v,ρ ∈ L^2([0,1] × T^d) by Cauchy-Schwarz, so the stronger L^{2+2ε} assertion is not needed for that step.","section":"Section 4, Proof of Theorem 1.7"},{"comment":"The claim that 'by (4.4), (4.7) and interpolation' one obtains v ∈ L^{2+2ε} is terse. One valid route is to interpolate each increment v_{q+1} − v_q between its L^2_{t,x} norm from (4.7) and its L∞_{t,x} bound inherited from (4.4), choosing ε < β/(4d+3), and then to sum the resulting geometric series; writing this out would clarify the argument.","section":"Section 4, Proof of Theorem 1.7"},{"comment":"The proof of Corollary 1.3 is given only by reference to [BCDL21, Theorem 1.3]. Since this is the deterministic Lagrangian analogue of the paper's main conclusion, a short sketch of the contradiction argument via the two measures obtained from the superposition principle and their time marginals would make the paper more self-contained.","section":"Section 2, Corollary 1.3"},{"comment":"It would be helpful to state explicitly that the martingale solutions produced by the superposition principle are converted into stochastic Lagrangian trajectories in the sense of Definition 1.4 via the martingale representation theorem for Brownian motion on the torus; currently this conversion is implicit.","section":"Section 1 and Section 4, Definitions 1.4–1.6 and Theorem 1.7"}],"recommendation":"minor_revision","confidential_remarks":"To the editor: the manuscript is technically sound in its main lines. The requested changes are clarifications of terse external-theorem steps and a precision issue about the exact Hölder exponent obtained from the iteration; none of these affects the central argument. The authors are transparent about the relation to the prior works [BSW23] and [Row24]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this deserves a serious referee. The new content is the simultaneous convex-integration construction for transport/Fokker-Planck and Euler/Navier-Stokes, giving sharp non-uniqueness for positive densities at the Onsager C^{1/3-} scale and for stochastic Lagrangian trajectories in the supercritical L^r_tL^p range. The paper is also honest about what is not new: it explicitly says plain transport non-uniqueness could be assembled from BSW23 and Row24. That kind of separation of new from prior is exactly what I want to see in a long construction paper.\n\nThe proofs are long but careful. I read through the parameter-heavy parts of Sections 3 and 5 and did not find a load-bearing gap. The reader's strongest worry about the \"without loss of generality, beta <= tilde beta\" reduction in Theorem 1.2(2) does not hold up. The line is terse, but it is justifiable: when the target tilde beta is below beta, choose any tilde beta0 in (beta, (1-beta)/2), run the iteration with the larger exponent, and the resulting density has more Holder regularity than needed. This should be spelled out in the final version, but it is not a gap.\n\nThe other flagged issue is also minor. In Theorem 1.7 the paper claims v is in L^{2+2epsilon}_{t,x} by interpolation from (4.4)-(4.7), and that claim is not obviously valid when p<2. But it is also unnecessary: the Trevisan superposition principle only needs the drift integrability condition on |v| with respect to the density, and that follows directly from the L^2_{t,x} bounds on v and rho. So the interpolation remark should be deleted or repaired, not the theorem.\n\nThe citation pattern looks healthy: the paper credits the external uniqueness results (Krylov-Rockner, Caravenna-Crippa) and the recent non-uniqueness inputs, and it does not oversell its own priority. My remaining reservation is just the usual one for this literature: the proof is extremely long, the parameter constraints are intricate, and there is no machine-checked verification. I did not find a specific error, but a result this dense needs a referee who will actually check the induction and the superposition-principle hypotheses against Trevisan's exact statements.\n\nWho is this for? Specialists in convex integration and stochastic fluid dynamics. It is not a casual read, but it is a genuine advance worth engaging with. My recommendation: send it to peer review, with a referee who knows the convex-integration literature and the Trevisan superposition principle.","headline":"A dense, honest convex-integration paper that reaches sharp non-uniqueness for deterministic and stochastic Lagrangian trajectories; the flagged weak spots are repairable and do not sink the main results.","tokens_in":92227,"tokens_out":3074,"would_cite":true,"duration_ms":34760,"reading_group":"yes","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":"For low-regularity Euler and Navier-Stokes solutions, particle trajectories need not be unique — even when a Brownian noise is added.","keywords":["Euler equations","Navier-Stokes equations","Lagrangian trajectories","stochastic Lagrangian trajectories","non-uniqueness in law","convex integration","transport equations","superposition principle"],"falsifier":"Inspect the two martingale solutions produced by the superposition principle from the density $\\rho$ of Theorem 1.10. If, for a positive-measure set of starting points, the regular conditional probabilities coincide, then the set $A(v)$ is empty and Theorem 1.7 fails; one could look for this by checking whether the constructed density $\\rho(t)$ differs from $1$ in $L^1$ at positive times, since identical trajectory laws would force $\\rho(t)=1$.","tokens_in":91143,"feed_emoji":"🌊","tokens_out":6759,"duration_ms":61101,"temperature":0.7,"pith_summary":"The paper proves that Lagrangian trajectories of weak solutions to the Euler and Navier-Stokes equations can be non-unique: there are divergence-free velocity fields, as regular as $C^{1/3-}$ in the 3D Euler case and $L^r_t L^p \\cap L^1_t W^{1,s}$ in general dimension, such that from a positive-measure set of starting points a particle has at least two possible futures. In the stochastic version, where the trajectory equation carries a Brownian noise, the two futures have genuinely different laws, so the noise does not automatically restore uniqueness. The construction works by running a convex-integration scheme on the fluid equation and the transported density simultaneously, producing two distinct solutions of the transport or Fokker-Planck equation from the same initial density. This matters because turbulence theory has long entertained the idea that in the high-Reynolds-number limit fluid particles become intrinsically random; the paper supplies deterministic fluids for which that randomness is provably forced.","feed_headline":"Weak Euler flows can give each particle multiple futures","feed_subtitle":"A convex-integration construction proves stochastic and deterministic Lagrangian trajectories fail uniqueness at supercritical regularity.","key_machinery":"The load-bearing object is a pair of convex-integration iterations run simultaneously: one builds the velocity field $v$ solving Euler or Navier-Stokes while the other builds a density $\\rho$ solving the transport or Fokker-Planck equation with the same $v$. At each step the approximations $(v_q,\\rho_q)$ come with error terms — the Reynolds stress $\\mathring{R}_q$ and the flux error $M_q$ — and the new perturbations are chosen from high-frequency building blocks, Mikado flows in the Hölder $C^0_{t,x}$ construction and generalized intermittent space-time jets in the $L^1_t W^{1,s}$ construction, whose supports are disjoint so that the perturbation cancelling $M_q$ does not destroy the cancellation of $\\mathring{R}_q$. Frequency parameters grow hypergeometrically and amplitudes are tuned so that both errors vanish in the limit, yielding a genuine solution pair. The density $\\rho$ is non-constant and stays positive while the constant density $1$ is also a solution; this is what forces non-uniqueness of the PDE, and afterwards of the trajectories.","core_discovery":"On the paper's own terms, the central discovery is: below the critical regularity thresholds, advection by a genuine weak solution of the incompressible Euler or Navier-Stokes equations does not determine a particle's fate. Theorem 1.2 gives 3D Euler solutions of Hölder regularity $\\beta<1/3$ with prescribed decreasing kinetic energy, for which the transport equation has a non-constant Hölder density starting from the constant state; since the constant density is always a solution, uniqueness fails at the PDE level, and via the superposition principle deterministic Lagrangian trajectories fail on a set of positive measure. Theorem 1.7 is the broader result: for $d\\geq 2$ and any supercritical triple $(p,r,s)$ in the set $A$, there is a divergence-free solution in $L^r_t L^p \\cap L^2_{t,x} \\cap L^1_t W^{1,s} \\cap C_t L^1$ to the Navier-Stokes or Euler equations for which the SDE $dX_t = v(t,X_t)\\,dt + \\sqrt{2\\kappa}\\,dW_t$ has, from a positive-measure set of starting points, two martingale solutions with distinct laws and finite expected drift cost. The paper also proves sharpness in the relevant directions: with time-continuous $L^p$ drifts for $p>2$ in two dimensions, or $W^{1,s}$ drifts with $s>d$, uniqueness is restored.","pith_inferences":["Beyond the paper: if the same two-scale iteration works with a nonlinear drift interaction or an external forcing, the mechanism suggests that non-uniqueness in law is generic for supercritical SDEs with divergence-free hydrodynamic drifts, not an isolated construction.","Beyond the paper: the positive-measure set $A(v)$ is obtained from regular conditional probabilities and could in principle be described explicitly from the convex-integration data; extracting its size or geometry would give quantitative information about how much of a particle cloud is genuinely stochastic.","Beyond the paper: a natural testable extension is to replace Brownian noise by fractional or Lévy noise; the Fokker-Planck equation would change its diffusion operator, and the superposition principle would have to be re-verified, so the method's reach beyond Brownian noise is open."],"forward_implications":["In three dimensions, dissipative Euler solutions with Hölder regularity below the Onsager exponent $1/3$ can have multiple deterministic Lagrangian trajectories from a positive-measure set of initial points.","In any dimension $d\\geq 2$, there are Navier-Stokes and Euler solutions in $L^r_t L^p$ with $1/p+1/r>1$ whose stochastic Lagrangian trajectories are non-unique in law, even though the drift is square-integrable in space-time.","For $p<2$, taking $r=\\infty$ gives time-continuous $C_t L^p$ drifts with non-unique deterministic trajectories, and for $\\kappa>0$ non-unique stochastic trajectories; these results are sharp unless the endpoint $p=2$ is added.","Known uniqueness results — stochastic uniqueness for $C_t L^p$ with $p>2$, deterministic uniqueness for $L^1_t W^{1,s}$ with $s>d$ — exactly delimit the thresholds, so the non-uniqueness is not an artifact of exotic regularity."],"supporting_citations":[{"why":"Supplies the superposition principle that turns the non-constant Fokker-Planck solution into two distinct stochastic trajectory laws.","marker":"[Tre14]"},{"why":"Provides the subcritical uniqueness result for stochastic flows, marking the sharp boundary Theorem 1.7 aims to reach.","marker":"[KR05]"},{"why":"Gives the deterministic uniqueness criterion for $W^{1,s}$ drifts with $s>d$ that delimits the sharp range in Corollary 1.9.","marker":"[CC21]"},{"why":"Provides the gluing and convex-integration scheme, including energy-profile control, adapted here to simultaneous transport equations.","marker":"[BDLSV19]"},{"why":"Introduces the gluing technique and proves Onsager's conjecture in the regularity regime the paper builds near.","marker":"[Ise18]"},{"why":"Supplies the building blocks and geometric lemmas for positive solutions of transport equations and non-unique trajectories.","marker":"[BCDL21]"},{"why":"Provides the commutator estimates used for the uniqueness half of Theorem 1.2.","marker":"[CET94]"}],"fun_headline_variants":["Weak Euler flows yield non-unique particle trajectories","Stochastic paths fail uniqueness in weak Euler solutions","Multiple particle paths exist in supercritical flows","Euler solutions allow multiple Lagrangian fates","Particle trajectories non-unique below critical regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction's bridge from PDE non-uniqueness to trajectory non-uniqueness is the superposition principle, an external result the paper applies without reproving; if that principle needs more regularity than the constructed drift and density actually have, Theorem 1.7 would not follow from Theorem 1.10.","fun_headline_variants_meta":{"raw":{"variants":["Weak Euler flows yield non-unique particle trajectories","Stochastic paths fail uniqueness in weak Euler solutions","Multiple particle paths exist in supercritical flows","Euler solutions allow multiple Lagrangian fates","Particle trajectories non-unique below critical regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000504,"raw_usage":{"total_tokens":2580,"prompt_tokens":1184,"completion_tokens":1396,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":800,"completion_tokens_details":{"reasoning_tokens":1327}},"tokens_in":800,"tokens_out":1396,"duration_ms":9068,"temperature":1.0,"reasoning_tokens":1327,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:58:27.412224+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Inspect the two martingale solutions produced by the superposition principle from the density $\\rho$ of Theorem 1.10. If, for a positive-measure set of starting points, the regular conditional probabilities coincide, then the set $A(v)$ is empty and Theorem 1.7 fails; one could look for this by checking whether the constructed density $\\rho(t)$ differs from $1$ in $L^1$ at positive times, since identical trajectory laws would force $\\rho(t)=1$.","supporting_citations":[],"review_version":1}