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

Huaxiang Lü, Michael Röckner, Xiangchan Zhu

TL;DR

The paper develops sharp non-uniqueness results for Lagrangian trajectories associated with Euler and Navier–Stokes flows by embedding a dual convex integration scheme that treats both the Euler equations and the transport (or Fokker–Planck) equations. On the $C_{t,x}^0$-scale, the authors construct dissipative Euler solutions with energy profiles dissipating above the Onsager threshold, and simultaneously build nontrivial densities solving transport equations to deduce non-uniqueness of deterministic Lagrangian trajectories via the superposition principle. On the $L^1_tW^{1,s}$-scale, they push into a supercritical regime where stochastic Lagrangian trajectories fail to be unique in law, using a two-scale convex integration with intensified intermittency and time jets to balance dissipative and advective effects. The results are sharp in 2D with known uniqueness under the LPS condition and in Sobolev classes with $s>d$, and extend the scope of spontaneous stochasticity and non-uniqueness phenomena for fluid models. Collectively, the work links Onsager-type energy dissipation, Lagrangian non-uniqueness, and stochastic regularization through a unified PDE and probabilistic framework, with implications for turbulence modeling and the mathematical foundations of fluid dynamics.

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}).

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

TL;DR

The paper develops sharp non-uniqueness results for Lagrangian trajectories associated with Euler and Navier–Stokes flows by embedding a dual convex integration scheme that treats both the Euler equations and the transport (or Fokker–Planck) equations. On the -scale, the authors construct dissipative Euler solutions with energy profiles dissipating above the Onsager threshold, and simultaneously build nontrivial densities solving transport equations to deduce non-uniqueness of deterministic Lagrangian trajectories via the superposition principle. On the -scale, they push into a supercritical regime where stochastic Lagrangian trajectories fail to be unique in law, using a two-scale convex integration with intensified intermittency and time jets to balance dissipative and advective effects. The results are sharp in 2D with known uniqueness under the LPS condition and in Sobolev classes with , and extend the scope of spontaneous stochasticity and non-uniqueness phenomena for fluid models. Collectively, the work links Onsager-type energy dissipation, Lagrangian non-uniqueness, and stochastic regularization through a unified PDE and probabilistic framework, with implications for turbulence modeling and the mathematical foundations of fluid dynamics.

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 regularity. As a corollary, in conjunction with the superposition principle, this yields the non-uniqueness of associated (deterministic) Lagrangian trajectories. Second, in dimension , for any or , we construct solutions to the Euler or Navier-Stokes equations in the space , 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 with , the associated stochastic Lagrangian trajectory associated with is unique (see \cite{KR05}); (2) in the deterministic case, the LPS condition guarantees that for any weak solution with to the Navier-Stokes equations, the associated (deterministic) Lagrangian trajectory is unique. Our result is also sharp in dimension in the sense that for any divergence-free vector field with , the associated (deterministic) Lagrangian trajectory is unique (see \cite{CC21}).

Paper Structure

This paper contains 28 sections, 19 theorems, 238 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convex integration and Onsager conjecture
  3. Previous results on the ODE level
  4. Previous results on the SDE level
  5. Ideas of the proof
  6. Ideas in the $C_{t,x}^0$-scales
  7. Ideas in the $L^1_tW^{1,s}$-scales
  8. Organization of the paper.
  9. Construction of non-unique solutions in $C_{t,x}^0$ scales
  10. Proof of Proposition \ref{['prop:1/3']}
  11. Choice of parameters and mollification
  12. The gluing procedure
  13. Gluing step for $(v_l,\mathring{R}_l)$
  14. Gluing step for $(\rho_l,M_l)$
  15. The perturbation procedure
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $\beta ,\tilde{\beta}\in(0, 1)$ and $T>0$ be fixed. (1). Let $\beta+2\tilde{\beta}>1$. For any divergence-free vector field $v\in C^\beta([0,T]\times \mathbb{T}^3)$, the transport equation eq:tpe has a unique solution $\rho\in C^{\tilde{\beta}}([0,T]\times \mathbb{T}^3)$, which conserves the kin such that there is a non-constant probability density $\rho \in C^{\tilde{\beta}}([0,T]\times \mat

Figures (1)

  • Figure 1: State of (non-)uniqueness to SDE \ref{['eq:sde']} for vector fields $v\in L^r_tL^p$ in the 2D case. Red area: in the subcritical case, the SDE admits a unique strong solution. Red line: in the critical case , the well-posedness of SDE remains open in $d=2$. Green area: in the supercritical case, our main result shows the non-uniqueness in law holds in this range.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 23 more