A very smooth ride in a rough sea
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}.
Keywords
Cite
@article{arxiv.1212.4333,
title = {A very smooth ride in a rough sea},
author = {U. Frisch and V. Zheligovsky},
journal= {arXiv preprint arXiv:1212.4333},
year = {2015}
}
Comments
8 pp., 19 refs. Communications in Mathematical Physics, in press