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Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

Published online by Cambridge University Press:  16 May 2014

Vladislav Zheligovsky  and
Uriel Frisch
Vladislav Zheligovsky
Affiliation:
Institute of Earthquake Prediction Theory and Mathematical Geophysics of the Russian Academy of Sciences, 84/32 Profsoyuznaya Street, 117997 Moscow, Russian Federation UNS, CNRS, Lab. Lagrange, OCA, CS 34229, 06304 Nice CEDEX 4, France
Uriel Frisch*
Affiliation:
UNS, CNRS, Lab. Lagrange, OCA, CS 34229, 06304 Nice CEDEX 4, France
*
†Email address for correspondence: uriel@obs-nice.fr
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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. To prove these 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 highly compressible flow, governed by the Euler–Poisson equations, a case of cosmological relevance. The recurrence relations associated with the Lagrangian formulation of these incompressible and compressible problems are so closely related that the proofs of time-analyticity are basically identical.

JFM classification

Mathematical Foundations: General fluid mechanics Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 749 , 25 June 2014 , pp. 404 - 430
Copyright
© 2014 Cambridge University Press 

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