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

We show that, for any Leray solution u to the 3D 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 [1] 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.