A very smooth ride in a rough sea
arXiv:1212.4333 · doi:10.1007/s00220-013-1848-1
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. Série 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,γ} Hölder norms of the coefficients and infer temporal analyticity of Lagrangian trajectories when the initial velocity is C^{1,γ}.
8 pp., 19 refs. Communications in Mathematical Physics, in press