SPASS - Probability, Stochastic Analysis and Statistics Seminar

A.e. uniqueness for (stochastic) Lagrangian trajectories for Leray solutions to 3D Navier-Stokes

by Lucio Galeati (EPFL)


 We revisit a result due to Robinson and Sadowski (2009), who first showed a.e. uniqueness of Lagrangian trajectories for admissible weak solutions to $3$D Navier-Stokes, for sufficiently regular $u_0$. We give an alternative proof, based on a newly established asymmetric Lusin-Lipschitz property of Leray solutions, exploited crucially in the arguments from Caravenna-Crippa (2021) and Brué-Colombo-De Lellis (2021). This approach is more robust, requiring no assumptions on $u_0$ and being applicable also to the stochastic characteristics of the system.
Finally, if $u_0$ is regular (say $u_0\in H^{1/2}$), then we are able to exploit the diffusive behaviour of stochastic trajectories to further prove that, for any fixed $x_0\in\mathbb{R}^d$, path-by-path uniqueness for the SDE $d X_t = u(t,X_t) d t + d B_t, X|t=0 = x_0$.