Regularity of Trajectories and Smooth Observables in Rough Euler Flows

For any Euler flow that is C^α uniformly in time, we show that all particle trajectories are of class C¹/(1-α) when 1/(1-α) is not an integer. This result holds despite the ill-posedness of the Euler equations and the expected nonuniqueness of trajectories. We discuss the implications for pair dispersion of particle trajectories. We also show that the Fourier coefficients and more generally the integral of velocity against any smooth function is C(1+α)/(1-α) in time when (1+α)/(1-α) is not an integer.