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

C J Cotter1, G A Gottwald2, D D Holm1

  • 1Department of Mathematics, Imperial College, London, UK.

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Summary

Stochastic fluid equations arise naturally from decomposing deterministic flow maps. This study derives effective stochastic particle dynamics using homogenization theory, validating previous findings.

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geometric mechanicshomogenizationmulti-scale fluid dynamicsstochastic fluid modelsstochastic processessymmetry reduced variational principles

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Area of Science:

  • Fluid dynamics
  • Stochastic processes
  • Homogenization theory

Background:

  • Previous work derived stochastic fluid equations using a variational principle and assumed stochastic Lagrangian particle dynamics.
  • The origin of these assumed dynamics within the deterministic fluid flow was not fully explained.

Purpose of the Study:

  • To demonstrate that stochastic Lagrangian dynamics naturally emerge from a multi-scale decomposition of deterministic fluid flow.
  • To derive effective stochastic particle dynamics and Eulerian stochastic fluid equations using homogenization theory.

Main Methods:

  • Multi-scale decomposition of the deterministic Lagrangian flow map into slow and fast components.
  • Application of homogenization theory to derive effective dynamics for the resolved mean part.
  • Assumption of mildly chaotic fast dynamics and a centring condition to ensure theoretical rigor.

Main Results:

  • The stochastic Lagrangian dynamics previously assumed are shown to naturally arise from the deterministic flow.
  • Effective slow stochastic particle dynamics are derived for the resolved mean flow.
  • Stochastic fluid partial equations in the Eulerian formulation are obtained.

Conclusions:

  • The study provides a rigorous derivation of stochastic fluid equations from deterministic fluid dynamics.
  • Homogenization theory offers a robust framework for understanding the emergence of stochasticity in fluid flows.
  • The findings validate and explain the underlying dynamics of previously assumed stochastic models.