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It is known that the Eulerian and Lagrangian structures of fluid flow can bedrastically different; for example, ideal fluid flow can have a trivial(static) Eulerian structure, while displaying chaotic streamlines. Here we showthat ideal flow with limited spatial smoothness (an initial vorticity that isjust a little better than continuous), nevertheless has time-analyticLagrangian trajectories before the initial limited smoothness is lost. Forproving such results we use a little-known Lagrangian formulation of idealfluid flow derived by Cauchy in 1815 in a manuscript submitted for a prize ofthe French Academy. This formulation leads to simple recurrence relations amongthe time-Taylor coefficients of the Lagrangian map from initial to currentfluid particle positions; the coefficients can then be bounded using elementarymethods. We first consider various classes of incompressible fluid flow,governed by the Euler equations, and then turn to a case of compressible flowof cosmological relevance, governed by the Euler-Poisson equations.

Time-analyticity of Lagrangian particle trajectories in ideal fluid flow