A very smooth ride in a rough sea

TL;DR

This paper provides an elementary derivation showing that in 3D incompressible Euler flows with initially smooth velocities, Lagrangian trajectories remain analytic in time over a finite period, based on Cauchy's form of the equations.

Contribution

It introduces a simple recurrence relation approach to prove temporal analyticity of Lagrangian trajectories for smooth initial velocities in Euler flows.

Findings

Lagrangian trajectories are analytic in time for smooth initial velocities.

Recurrence relations among Taylor coefficients are derived and used.

Bounds on Hölder norms of coefficients are established.

Abstract

It has been known for some time that a 3D incompressible Euler flow that has initially a barely smooth velocity field nonetheless has Lagrangian fluid particle trajectories that are analytic in time for at least a finite time (Ph. Serfati C.R. Acad. Sci. S\'erie I 320, 175-180 (1995); A. Shnirelman arXiv:1205.5837 (2012)). Here an elementary derivation is given, based on Cauchy's form of the Euler equations in Lagrangian coordinates. This form implies simple recurrence relations among the time-Taylor coefficients of the Lagrangian map, used here to derive bounds for the C^{1,\gamma} H\"older norms of the coefficients and infer temporal analyticity of Lagrangian trajectories when the initial velocity is C^{1,\gamma}.