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A Very Smooth Ride in a Rough Sea

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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 (C. R. Acad. Sci. Paris Série I 320:175–180, 1995), Shnirelman (Glob. Stoch. Anal., http://arxiv.org/abs/1205.5837v1, 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.

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Authors and Affiliations

  1. UNS, CNRS, Lab. Lagrange, OCA, B.P. 4229, 06304, Nice Cedex 4, France

    Uriel Frisch & Vladislav Zheligovsky

  2. Institute of Earthquake Prediction Theory and Mathematical Geophysics of the Russ. Ac. Sci., 84/32 Profsoyuznaya St, 117997, Moscow, Russia

    Vladislav Zheligovsky

Authors
  1. Uriel Frisch
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  2. Vladislav Zheligovsky
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Correspondence to Uriel Frisch.

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Communicated by L. Caffarelli

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Frisch, U., Zheligovsky, V. A Very Smooth Ride in a Rough Sea. Commun. Math. Phys. 326, 499–505 (2014). https://doi.org/10.1007/s00220-013-1848-1

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  • DOI: https://doi.org/10.1007/s00220-013-1848-1

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