A very smooth ride in rough sea

TYPEAstrophysics Seminar
Speaker:Professor Uriel Frisch
Affiliation:Observatoire de la Cote d'Azur, Nice, France
Time:10:30 - 11:30
Location:Lewiner Seminar Room (412)
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 (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.

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.

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

This talk is based in part on a paper with V. Zheligovsky, to appear
in Comm. Math. Phys., arXiv:1212.4333 [math.AP].