A very smooth ride in rough sea
Observatoire de la Cote d'Azur, Nice, France
Lundi 17/06/2013, 11:30
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
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 (Serfati, 1995; Shnirelman, 2012). Here, an elementary derivation is given, based on a little-known form of the Euler equations in Lagrangian coordinates, discovered by Cauchy in 1815. par This form implies simple recurrence relations among the time-Taylor coefficients of the Lagrangian map, used here to derive bounds for the Hoelder norms of the Lagrangian gradients of the Taylor coefficients and infer temporal analyticity of Lagrangian trajectories when the initial vorticity is itself Hoelder continuous. par The same kind of proof holds for the temporal analyticity of Lagrangian trajectories in an Einstein-de Sitter Universe governed by the Euler-Poisson equations, provided the so-called linear growth factor of density is used as time variable. Actually, the Lagrangian perturbation expansion, introduced by cosmologists in the 90s (Buchert, Moutarde et al.), is then basically a temporal Taylor expansion.