Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

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