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

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.