Random Splitting of Fluid Models: Ergodicity and Convergence

TL;DR

This paper introduces stochastic fluid models that decompose deterministic dynamics into fundamental components, inject randomness, and demonstrate ergodicity and convergence, with applications to Lorenz-96 and fluid equations.

Contribution

It presents a novel stochastic modeling framework for fluid dynamics that preserves key properties and proves convergence to deterministic models under general conditions.

Findings

Unique invariant measure established for the models

Models converge almost surely to deterministic dynamics in small noise limit

Applicable to conservative fluid models like Euler and Navier-Stokes

Abstract

We introduce a family of stochastic models motivated by the study of nonequilibrium steady states of fluid equations. These models decompose the deterministic dynamics of interest into fundamental building blocks, i.e., minimal vector fields preserving some fundamental aspects of the original dynamics. Randomness is injected by sequentially following each vector field for a random amount of time. We show under general assumptions that these random dynamics possess a unique invariant measure and converge almost surely to the original, deterministic model in the small noise limit. We apply our construction to the Lorenz-96 equations, often used in studies of chaos and data assimilation, and Galerkin approximations of the 2D Euler and Navier-Stokes equations. An interesting feature of the models developed is that they apply directly to the conservative dynamics and not just those with excitation and dissipation.