Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations
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arXiv
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2025
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| author | Lü, Huaxiang Röckner, Michael Zhu, Xiangchan |
| author_facet | Lü, Huaxiang Röckner, Michael Zhu, Xiangchan |
| contents | 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}). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16687 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations Lü, Huaxiang Röckner, Michael Zhu, Xiangchan Analysis of PDEs Mathematical Physics Probability 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}). |
| title | Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations |
| topic | Analysis of PDEs Mathematical Physics Probability |
| url | https://arxiv.org/abs/2504.16687 |