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

TL;DR

This paper proves the almost-everywhere uniqueness of Lagrangian trajectories for 3D Navier--Stokes solutions, extending previous results and employing new techniques involving Lusin--Lipschitz properties.

Contribution

It establishes almost-everywhere and pathwise uniqueness of Lagrangian trajectories for Leray solutions, improving and extending prior work with novel methods.

Findings

Uniqueness of trajectories for Leray solutions with initial data in L^2.

Pathwise uniqueness for initial data in H^{1/2}.

Introduction of a new asymmetric Lusin--Lipschitz property for Leray solutions.

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.