Uniqueness of the particle trajectories of the weak solutions of the two - dimensional Navier - Stokes equations ⋆

We address the problem of the uniqueness of the particle trajectories corresponding to a weak solution of the two-dimensional Navier-Stokes equations with square integrable initial condition and regular enough forcing function. In order to do this we obtain an estimate for H-norm of the weak solutions of the Navier-Stokes equations and show that t‖D3/2u‖ is bounded which provides a bound on the time derivative of the corresponding trajectories near the initial time. This, combined with an appropriate log-Lipschitz bound on the velocity field, gives the uniqueness. We also show that the same approach can be used to prove the uniqueness of the trajectories corresponding to the local strong solutions of the three-dimensional Navier-Stokes equations when the H-norm of the initial condition is finite.