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Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics

Presented by: 
Darryl Holm Imperial College London
Date: 
Friday 22nd September 2017 - 10:20 to 11:00
Venue: 
INI Seminar Room 1
Abstract: 
Co-authors: Colin J Cotter (Imperial College London), Georg A Gottwald (University of Sydney)

In [Holm, Proc. Roy. Soc. A 471 (2015)] stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby justifying stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centering condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.

Joint work with Colin J Cotter (Imperial College London) Georg A Gottwald (University of Sydney).
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons