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Almost-everywhere uniqueness of Lagrangian trajectories for $3$D Navier--Stokes revisited

Lucio Galeati

TL;DR

This work proves almost-everywhere and pathwise uniqueness results for Lagrangian trajectories associated with 3D Navier–Stokes Leray solutions. By developing a pathwise theory of Random Regular Lagrangian Flows and establishing an asymmetric Lusin–Lipschitz property via refined Lorentz and Besov interpolation, the authors obtain a.e. trajectorial uniqueness for deterministic and stochastic characteristics, and pathwise uniqueness for SDEs with additive noise under minimal regularity; stronger data ($u_0\in H^{1/2}$) yields path-by-path uniqueness for all initial conditions. The results extend Robinson–Sadowski (2009) to a broader setting with forcing terms and Brownian noise, and provide a robust framework to analyze Lagrangian trajectories under low-regularity NS dynamics. The work also derives higher-order Leray regularity estimates and shows how these feed the RLF machinery, yielding new insights into the well-posedness of stochastic characteristics and semigroup properties for the corresponding advection–diffusion systems. Overall, the paper bridges nonlinear PDE regularity with stochastic characteristics, offering quantitative convergence for Picard iterations and laying groundwork toward potential Constantin–Iyer representations in this low-regularity regime.

Abstract

We show that, for any Leray solution $u$ to the $3$D Navier--Stokes equations with $u_0\in L^2$, the associated deterministic and stochastic Lagrangian trajectories are unique for Lebesgue a.e. initial condition. Additionally, if $u_0\in H^{1/2}$, then pathwise uniqueness is established for the stochastic Lagrangian trajectories starting from every initial condition. The result sharpens and extends the original one by Robinson and Sadowski (Nonlinearity 2009) and is based on rather different techniques. A key role is played by a newly established asymmetric Lusin--Lipschitz property of Leray solutions $u$, in the framework of (random) Regular Lagrangian flows.

Almost-everywhere uniqueness of Lagrangian trajectories for $3$D Navier--Stokes revisited

TL;DR

This work proves almost-everywhere and pathwise uniqueness results for Lagrangian trajectories associated with 3D Navier–Stokes Leray solutions. By developing a pathwise theory of Random Regular Lagrangian Flows and establishing an asymmetric Lusin–Lipschitz property via refined Lorentz and Besov interpolation, the authors obtain a.e. trajectorial uniqueness for deterministic and stochastic characteristics, and pathwise uniqueness for SDEs with additive noise under minimal regularity; stronger data () yields path-by-path uniqueness for all initial conditions. The results extend Robinson–Sadowski (2009) to a broader setting with forcing terms and Brownian noise, and provide a robust framework to analyze Lagrangian trajectories under low-regularity NS dynamics. The work also derives higher-order Leray regularity estimates and shows how these feed the RLF machinery, yielding new insights into the well-posedness of stochastic characteristics and semigroup properties for the corresponding advection–diffusion systems. Overall, the paper bridges nonlinear PDE regularity with stochastic characteristics, offering quantitative convergence for Picard iterations and laying groundwork toward potential Constantin–Iyer representations in this low-regularity regime.

Abstract

We show that, for any Leray solution to the D Navier--Stokes equations with , the associated deterministic and stochastic Lagrangian trajectories are unique for Lebesgue a.e. initial condition. Additionally, if , then pathwise uniqueness is established for the stochastic Lagrangian trajectories starting from every initial condition. The result sharpens and extends the original one by Robinson and Sadowski (Nonlinearity 2009) and is based on rather different techniques. A key role is played by a newly established asymmetric Lusin--Lipschitz property of Leray solutions , in the framework of (random) Regular Lagrangian flows.
Paper Structure (13 sections, 22 theorems, 135 equations)

This paper contains 13 sections, 22 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Main results and ideas of proof
  3. (Random) Regular Lagrangian flows and a.e. uniqueness
  4. (Random) Regular Lagrangian flows
  5. Asymmetric Lusin--Lipschitz property and a.e. trajectorial uniqueness
  6. Quantitative convergence of Picard iterations
  7. Interpolation-type inequalities in Lorentz and Besov spaces
  8. Inequalities in Lorentz spaces
  9. Inequalities in Besov spaces
  10. Applications to $3$D Navier--Stokes
  11. Regularity estimates for Leray solutions
  12. Proof of the main results
  13. Consequences

Key Result

Theorem 1.2

Let $u_0\in L^2_\sigma$ and $u$ be an associated Leray solution. Then the ODE problem is well-defined and for Lebesgue a.e. $x\in{\mathbb R}^3$ there exists exactly one solution to the above ODE.

Theorems & Definitions (60)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 50 more