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2013

Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

Zheligovsky, Vladislav, Frisch, Uriel

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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.

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