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

Dashti, M and Robinson, J C (2009) A simple proof of uniqueness of the particle trajectories for solutions of the Navier-Stokes equations. Nonlinearity, 22 (4). pp. 735-746. ISSN 0951-7715

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We give a simple proof of the uniqueness of fluid particle trajectories corresponding to (1) the solution of the two-dimensional NavierStokes 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 and Lerner (1995 J. Diff. Eqns 121 31428) using the LittlewoodPaley theory for the flow in the whole space $\mathbb{R}^d$ , $d \ge 2$. 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 $p &gt; 1$ 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 NavierStokes 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.