A simple proof of uniqueness of the particle trajectories for solutions of the Navier-Stokes equations

TL;DR

This paper presents a straightforward proof of the uniqueness of particle trajectories in 2D and 3D Navier-Stokes solutions with minimal regularity assumptions, simplifying previous complex methods.

Contribution

It introduces a simplified proof of trajectory uniqueness for Navier-Stokes solutions with minimal Sobolev regularity, extending to bounded domains.

Findings

Uniqueness of particle trajectories under minimal regularity conditions.

Proof applicable to bounded domains.

Simplifies previous proofs using energy methods.

Abstract

We give a simple proof of the uniqueness of fluid particle trajectories corresponding to: 1) the solution of the two-dimensional Navier Stokes equations with an initial condition that is only square integrable, and 2) the local strong solution of the three-dimensional equations with an $H^{1/2}$-regular initial condition i.e.\ with the minimal Sobolev regularity known to guarantee uniqueness. This result was proved by Chemin & Lerner (J Diff Eq 121 (1995) 314-328) using the Littlewood-Paley theory for the flow in the whole space $\R^d$, $d\ge 2$. We first show that the solutions of the differential equation $\dot{X}=u(X,t)$ are unique if $u\in L^p(0,T;H^{(d/2)-1})$ for some $p>1$ and $\sqrt{t}\,u\in L^2(0,T;H^{(d/2)+1})$. We then prove, using standard energy methods, that the solution of the Navier-Stokes equations with initial condition in $H^{(d/2)-1}$ satisfies these conditions. This proof is also valid for the more physically relevant case of bounded domains.