Endpoint cases of the sharp uniqueness threshold for (stochastic) Lagrangian trajectories of Euler and Navier-Stokes flows

Lü, Röckner and Zhu (arXiv:2504.16687; Journal of Functional Analysis, 2026) constructed, by a simultaneous convex integration scheme for the incompressible Euler equations and an advected density, weak solutions whose (stochastic) Lagrangian trajectories are non-unique in law or as classical characteristics, sharply in three regularity scales but leaving the endpoint cases open. The open problem is to complete the sharp uniqueness threshold by deciding each endpoint regime: (1) the joint Hölder line $\beta+2\tilde\beta=1$ for the transport equation advected by Onsager-critical 3D Euler flows, where uniqueness and energy conservation hold for $\beta+2\tilde\beta>1$ and non-uniqueness holds for $\beta+2\tilde\beta<1$; (2) the critical Ladyzhenskaya-Prodi-Serrin boundary $d/p+2/r=1$ for well-posedness in law of the associated stochastic differential equations, in dimension $d=2$ including the endpoint $p=2$ of the time-continuous scale $C_tL^p$; and (3) the Sobolev endpoint $s=d$ for almost-everywhere uniqueness of deterministic characteristics. Each of these regimes must be resolved either by a uniqueness theorem in the natural solution class or by a non-uniqueness construction attaining the critical exponent.

Background

In the Lagrangian description of an incompressible fluid, a particle starting at position $x$ at time $0$ moves along a trajectory solving the ordinary differential equation $\dot X(t)=v(t,X(t))$, $X(0)=x$ (or, with external noise, the stochastic differential equation $dX_t=v(t,X_t)\,dt+\sqrt{2\kappa}\,dW_t$ with $W_t$ a standard Brownian motion and $\kappa\ge 0$), where the velocity field $v$ is a weak solution of the incompressible Euler equations (viscosity $\nu=0$) or Navier-Stokes equations ($\nu>0$) on the torus $\mathbb T^d$. For smooth $v$ the trajectory is unique and depends stably on the initial point, but weak solutions of the fluid equations can have very low regularity, and for such velocities both uniqueness and well-posedness in law of the trajectories can fail.

Two classical frameworks quantify how much regularity of $v$ forces uniqueness. The DiPerna-Lions theory shows that a divergence-free velocity $v\in L^1_tW^{1,s}$ generates a unique regular Lagrangian flow (a measure-preserving flow respecting the continuity equation); Caravenna and Crippa strengthened this to almost-everywhere uniqueness of individual trajectories when $s>d$. Conversely, Bruè, Colombo and De Lellis constructed divergence-free velocities $v\in C_t(W^{1,d-}\cap L^{\infty-})$ whose characteristic curves are genuinely non-unique on a set of positive measure, Kumar extended such non-uniqueness to a set of initial data of full measure below the Sobolev threshold, and Bruè, Colombo and Kumar identified the sharp density-class threshold $1/p+(d-1)/(dr)=1$ for $L^\infty_tW^{1,p}$ velocities. The Sobolev-scale threshold thus sits at the endpoint $s=d$, which remains undecided.

On the stochastic side, Krylov and Röckner proved strong existence and uniqueness for the stochastic differential equation with drift $v\in L^r_tL^p$ under $d/p+2/r<1$, the stochastic analogue of the Ladyzhenskaya-Prodi-Serrin (LPS) condition $d/p+2/r\le 1$ from Navier-Stokes theory; in particular, in $d=2$ time-continuous divergence-free velocities $v\in C_tL^p$ with $p>2$ generate unique trajectories in law. At criticality, Krylov proved strong well-posedness for time-independent drifts in $L^d$, and Röckner and Zhao proved strong and weak well-posedness in the critical classes $C_tL^d$ and $L^r_tL^p$ with $r,p\in(2,\infty)$, and weak uniqueness for divergence-free $L^\infty_tL^d$ drifts. Each of these critical results, however, operates within a restricted class of solutions (those satisfying a Krylov-type estimate), and the borderline — especially the whole critical LPS line in $d=2$, including the endpoint $(p,r)=(2,\infty)$ — remains open.

In the Hölder scale, Onsager's conjecture asserts that $1/3$ is the critical Hölder exponent for energy conservation of Euler solutions: the rigid side was proved by Constantin, E and Titi via commutator estimates, and the flexible side by convex integration (Buckmaster, De Lellis, Székelyhidi Jr. and Vicol). Convex integration is an iterative perturbative technique that constructs weak solutions with prescribed anomalous features by adding finely oscillatory correctors that cancel an error term at each step; it is also the engine of all known non-uniqueness constructions at these regularities. For the transport equation $\partial_t\rho+\operatorname{div}(v\rho)=0$, commutator estimates of Constantin-E-Titi type give uniqueness of density solutions $\rho\in C^{\tilde\beta}$ for $v\in C^\beta$ whenever $\beta+2\tilde\beta>1$, and the same quantity is the correct joint scale: the anomalous-dissipation constructions of Burczak, Székelyhidi Jr. and Wu for Euler flows in $C^{1/3-}$, combined with the result of Rowan that anomalous dissipation of the scalar implies non-uniqueness of the associated (stochastic) trajectories, produce non-uniqueness strictly below the line $\beta+2\tilde\beta=1$.

The Eulerian and Lagrangian pictures are tied together by the superposition principle: non-uniqueness of probability-density solutions of the transport (or Fokker-Planck) equation with a fixed initial datum is equivalent to non-uniqueness in law of the associated (stochastic) trajectories. This is why constructing a non-constant density $\rho$ with initial datum $\rho_0\equiv 1$ — the constant $1$ being always a solution for divergence-free $v$ — yields two distinct trajectory laws started from the invariant initial distribution.

Lü, Röckner and Zhu ran a simultaneous convex integration for the coupled system (Euler equation plus advected density) and proved: (a) (Theorem 1.2) in $d=3$, for $0<\beta+2\tilde\beta<1$ with $0<\beta<1/3$, there exists a weak solution $v\in C^\beta$ of the Euler equations with prescribed kinetic energy profile and a non-constant probability density $\rho\in C^{\tilde\beta}([0,T]\times\mathbb T^3)$ solving the transport equation with $\rho_0=1$; (b) (Theorem 1.7 and Corollaries 1.8, 1.9) in $d\ge 2$, for supercritical Lebesgue and Sobolev exponents — in particular Euler/Navier-Stokes solutions $v\in C_tL^p$ for every $1<p<2$ in $d=2$ (with $\kappa\in[0,1]$ arbitrary, so both stochastic and deterministic trajectories are covered) and $v\in L^1_tW^{1,s}$ for every $s<d$ — there exist solutions whose stochastic Lagrangian trajectories are non-unique in law, even when started from the invariant density $\rho\equiv 1$. On the temporal regularity of the advected densities, Remark 2.2 of the source records that the direct BDLSV19-style argument yields only $\tilde\beta/(1+\tilde\beta-\beta)<\tilde\beta$, while the refined scheme of the paper reaches every exponent strictly below $\tilde\beta$. All three scales are sharp against the corresponding uniqueness results, except at the endpoints, which the authors leave open and which constitute the present problem.

Problem Statement

Determine the sharp uniqueness/non-uniqueness threshold for (stochastic) Lagrangian trajectories associated with weak solutions of the incompressible Euler and Navier-Stokes equations, by resolving each endpoint regime left open between the known uniqueness results and the constructions of Lü-Röckner-Zhu:

(1) Hölder/Onsager endpoint (transport level, $d=3$). On the joint Hölder line $\beta+2\tilde\beta=1$ with $0<\beta<1/3$ and $\tilde\beta\in(0,1)$: either prove that for every divergence-free $v\in C^\beta([0,T]\times\mathbb T^3)$ the transport equation $\partial_t\rho+\operatorname{div}(v\rho)=0$ with $\rho|_{t=0}=1$ has a unique solution in the natural Hölder class $C^{\tilde\beta}$ (with the corresponding conservation of the density's $L^2$ norm, extending the commutator argument to the endpoint), or construct a weak solution $v\in C^\beta$ of the 3D Euler equations — with the usual freedom to prescribe a strictly positive smooth kinetic energy profile — together with a non-constant probability density $\rho\in C^{\tilde\beta}([0,T]\times\mathbb T^3)$ solving the same initial-value problem (giving two solutions together with $\bar\rho\equiv 1$, and hence non-unique deterministic Lagrangian trajectories via the superposition principle).

(2) Critical LPS boundary (stochastic trajectories, $d=2$). (a) At the endpoint $(p,r)=(2,\infty)$ of the time-continuous scale: decide, for both $\kappa>0$ (stochastic trajectories, uniqueness in law) and $\kappa=0$ (deterministic characteristics, almost-everywhere uniqueness, for Leray weak solutions of the Navier-Stokes equations, mirroring the deterministic comparison in the source for $C_tL^p$ solutions with $p>2$), whether uniqueness holds for every divergence-free Euler/Navier-Stokes solution $v\in C_tL^2(\mathbb T^2)$ — or whether non-uniqueness persists at $p=2$ for suitable solutions, e.g. with two distinct trajectory laws started from the invariant density $\rho\equiv 1$. (b) More generally, on the critical LPS line $1/p+1/r=1/2$ in $d=2$: decide well-posedness in law for divergence-free drifts $v\in L^r_tL^p(\mathbb T^2)$ beyond the sub-ranges and solution-class restrictions (Krylov-type estimate assumptions) of the known critical results of Krylov and Röckner-Zhao — either remove those restrictions or exhibit a critical counterexample.

(3) Sobolev endpoint (deterministic trajectories, $d\ge 2$). At $s=d$: either prove almost-everywhere uniqueness of characteristics for every divergence-free $v\in L^1_tW^{1,d}(\mathbb T^d)$ (extending the Caravenna-Crippa theory to the endpoint), or construct an Euler/Navier-Stokes solution $v\in L^1_tW^{1,d}(\mathbb T^d)$ (retaining the side integrability used in the known constructions, such as $v\in L^2_{t,x}$) whose deterministic Lagrangian trajectories are non-unique on a set of initial data of positive measure.

Components (1)-(3) are the critical cases of one and the same dichotomy — the exact regularity at which uniqueness of trajectories is lost on each of the three standard scales (Hölder/Onsager, Lebesgue/LPS, Sobolev); of these, (1), (2a) and (3) are single endpoints, while (2b) is the one-parameter critical LPS line. A resolution of any single sub-question on either side (a uniqueness theorem or a non-uniqueness construction) constitutes genuine progress; the problem is fully resolved when each of (1), (2a), (2b) and (3) has been decided.

References

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