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

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Title: Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics
Authors: Cotter, CJ
Gottwald, G
Holm, DD
Item Type: Journal Article
Abstract: In Holm (Holm 2015 Proc. R. Soc. A 471, 20140963. (doi:10.1098/rspa.2014.0963)), 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 obtaining 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 centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.
Issue Date: 30-Sep-2017
Date of Acceptance: 17-Aug-2017
URI: http://hdl.handle.net/10044/1/50622
DOI: 10.1098/rspa.2017.0388
ISSN: 1364-5021
Publisher: Royal Society of London
Journal / Book Title: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume: 473
Issue: 2205
Copyright Statement: © 2017 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Natural Environment Research Council (NERC)
Funder's Grant Number: EP/N023781/1
EP/L000407/1
NE/G000212/1
Keywords: Science & Technology
Multidisciplinary Sciences
Science & Technology - Other Topics
geometric mechanics
stochastic fluid models
stochastic processes
multi-scale fluid dynamics
symmetry reduced variational principles
homogenization
NONUNIFORMLY HYPERBOLIC SYSTEMS
SURE INVARIANCE-PRINCIPLE
FLOW
geometric mechanics
homogenization
multi-scale fluid dynamics
stochastic fluid models
stochastic processes
symmetry reduced variational principles
math.AP
math.AP
math-ph
math.DS
math.MP
physics.flu-dyn
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status: Published
Open Access location: https://arxiv.org/abs/1706.00287
Article Number: 20170388
Online Publication Date: 2017-09-20
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences