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

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.