Circulation and energy theorem preserving stochastic fluids

Theodore D. Drivas, Darryl D. Holm

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2776-2814
Number of pages39
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Issue number6
StatePublished - Dec 2020

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Kelvin theorem
  • stochastic fluid equations
  • variational principle


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