Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations

Huaxiang L\"u; Michael R\"ockner; Xiangchan Zhu

Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations

T0 review · 0 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For low-regularity Euler and Navier-Stokes solutions, particle trajectories need not be unique — even when a Brownian noise is added.

desk verdict 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. read the letter →

arxiv 2504.16687 v2 pith:7BR3IXLH submitted 2025-04-23 math.AP math-phmath.MPmath.PR

Huaxiang Lü , Michael Röckner , Xiangchan Zhu This is my paper · ORCID

classification math.APmath-phmath.MPmath.PR MSC 60H1535R6035Q30

keywords EulerequationsNavier-StokesLagrangiantrajectoriesstochasticnon-uniquenessinlawconvexintegrationtransportsuperpositionprinciple

topics Navier-Stokes Regularity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $\rho(t)=1$.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

The 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.

Who 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.

Referee Report

0 major / 6 minor

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.

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.

minor comments (6)

[Section 2, Proof of Theorem 1.2] 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 2, Proposition 2.1 and its hypotheses] 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 4, Proof of Theorem 1.7] 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 4, Proof of Theorem 1.7] 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 2, Corollary 1.3] 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 1 and Section 4, Definitions 1.4–1.6 and Theorem 1.7] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convex-integration construction is self-contained, and the superposition-principle and sharpness inputs are external theorems with independently verifiable hypotheses.

full rationale

The paper's central claims are produced by explicit convex-integration iterations rather than by fitting or by assuming the conclusion. Theorem 1.2(2) is proved through Proposition 2.1 with a full inductive construction of (v_q, ρ_q, stress terms) in Sections 2-3, including parameter choices, gluing steps, perturbation amplitudes, stress estimates, and energy checks. Theorem 1.10, and hence Theorem 1.7, is proved through Proposition 4.1 in Sections 4-5 with its own explicit generalized intermittent jets and temporal correctors. The passage from the non-constant Fokker-Planck solution ρ to distinct trajectory laws invokes the superposition principle of Trevisan [Tre14, Section 7.2] as an external theorem; the paper verifies the needed integrability (∫∫|v|ρ<∞ follows immediately from v,ρ∈L^2_{t,x} by Cauchy-Schwarz, and weak continuity follows from ρ∈C([0,T];L^1)), so this step is not a relabeled input. The claim that v∈L^{2+2ε} follows by interpolation from (4.4)-(4.7) is not clearly justified for p<2, but that stronger integrability is not needed for the superposition principle, so this is at most a minor correctness blemish, not circularity. Sharpness statements cite independent external uniqueness theorems: Krylov-Röckner [KR05] for stochastic trajectories with p>2, Caravenna-Crippa [CC21] for deterministic trajectories in W^{1,s}, s>d, and the Ladyzhenskaya-Prodi-Serrin theory for Navier-Stokes. Although one author is a coauthor of [KR05], that theorem is an independent published result whose hypotheses do not include the present conclusion, and it is used only to demarcate the sharp range, not to construct the counterexample. The 'without loss of generality β≤β̃' reduction in Section 2 is also legitimate by monotonicity of the exponent constraints. Thus every load-bearing input is either proved in the paper or is an external theorem with stated, checkable hypotheses; no equation is simultaneously a hypothesis and a conclusion.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central construction is a standard convex integration proof with many internal parameters; none of them are fitted to external data, and no new physical entities are postulated. The sharpness statements rely on external uniqueness theorems, while the main existence arguments are carried out inside the paper.

free parameters (5)

frequency growth base a
A large positive real chosen to absorb universal constants and to make the iteration inequalities hold. It is a proof parameter, not fitted to data.
frequency growth exponent b
Chosen close to 1 in Section 2 and large in Section 4 to satisfy inequalities such as (3.1) and (5.1). Determines the hypergeometric growth of the oscillation frequencies.
mollification length l
Defined in Section 3.1 and Section 5.1 as a ratio of amplitude parameters and frequencies; controls all derivative losses in the convex integration iteration.
gluing interval length tau_q
Chosen in Section 3.2 to satisfy the CFL-type condition (3.9), ensuring the local gluing solutions are well defined and the residual stresses are localized on disjoint time intervals.
intermittency parameters r_perp, r_parallel, eta, sigma, mu
Chosen in Lemma 5.1 and equation (5.10) to satisfy (5.8) and (5.9). These set the spatial and temporal concentration scales of the intermittent jets used in the L^1_tW^{1,s} construction.

assumptions (5)

domain assumption Superposition principle for Fokker-Planck equations with low-regularity drift, from Trevisan [Tre14, Section 7.2].
Used in the proof of Theorem 1.7 to pass from non-unique densities to non-unique martingale solutions for the SDE. This external theorem is not reproved, and the paper only checks the necessary integrability and weak continuity.
domain assumption Krylov-Rockner strong well-posedness for stochastic differential equations with d/p + 2/r < 1.
Used to claim sharpness of the stochastic non-uniqueness result for p > 2 with r = infinity. This is an external uniqueness theorem.
domain assumption Caravenna-Crippa uniqueness of regular Lagrangian flows for divergence-free v in L^1_tW^{1,s} with s > d.
Used to claim sharpness of the deterministic non-uniqueness result. Cited as [CC21, Corollary 5.2].
domain assumption Ladyzhenskaya-Prodi-Serrin regularity for Navier-Stokes weak solutions in C_tL^p with p > d.
Used to argue that deterministic Lagrangian trajectories are unique for such regular solutions. This is an external theorem invoked in the introduction.
standard math Standard Calderon-Zygmund, commutator, and transport estimates for the inverse divergence operators on the torus.
Proved or cited in Appendix A and used throughout the convex integration constructions. These are standard analytic tools, not specific to the paper's claims.

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Pith. "Pith review of Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations." pith.science (2026). https://pith.science/paper/7BR3IXLH

@misc{pith2026250416687,
  author       = {Pith},
  title        = {Pith review of: Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BR3IXLH}},
  note         = {Machine review of arXiv:2504.16687}
}

abstract

We are concerned with the (stochastic) Lagrangian trajectories associated with Euler or Navier-Stokes equations. First, in the vanishing viscosity limit, we establish sharp non-uniqueness results for positive solutions to transport equations advected by weak solutions of the 3D Euler equations that exhibit kinetic energy dissipation with $C_{t,x}^{1/3-}$ regularity. As a corollary, in conjunction with the superposition principle, this yields the non-uniqueness of associated (deterministic) Lagrangian trajectories. Second, in dimension $d\geq2$, for any $\frac{1}{p}+\frac{1}{r}>1$ or $p\in(1,2),r=\infty$, we construct solutions to the Euler or Navier-Stokes equations in the space $L_t^rL^p\cap L_t^1W^{1,1}$, demonstrating that the associated (stochastic) Lagrangian trajectories are not unique. Our result is sharp in 2D in the sense that: (1) in the stochastic case, for any vector field $v\in C_tL^p$ with $p>2$, the associated stochastic Lagrangian trajectory associated with $v$ is unique (see \cite{KR05}); (2) in the deterministic case, the LPS condition guarantees that for any weak solution $v\in C_tL^p$ with $p>2$ to the Navier-Stokes equations, the associated (deterministic) Lagrangian trajectory is unique. Our result is also sharp in dimension $d\geq2$ in the sense that for any divergence-free vector field $v\in L_t^1W^{1,s}$ with $s>d$, the associated (deterministic) Lagrangian trajectory is unique (see \cite{CC21}).

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