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Circulation and Energy Theorem Preserving Stochastic Fluids

Published online by Cambridge University Press:  23 July 2019

Theodore D. Drivas  and
Darryl D. Holm
Theodore D. Drivas
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, United States (tdrivas@math.princeton.edu)
Darryl D. Holm
Affiliation:
Department of Mathematics, Imperial College, London SW7 2AZ, UK (d.holm@imperial.ac.uk)
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Abstract

Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

Keywords

stochastic fluid equationsvariational principleKelvin theorem

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Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2776 - 2814
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Copyright © 2019 The Royal Society of Edinburgh

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