Stochastic stability of invariant measures: The 2D Euler equation

TL;DR

This paper explores the stochastic stability of invariant measures in the 2D Euler equation, proposing that observed coherent structures are linked to stochastically stable measures influenced by boundary conditions.

Contribution

It extends stochastic stability concepts from finite to infinite-dimensional systems, specifically analyzing the 2D Euler equation and its invariant measures.

Findings

Coherent structures correspond to stochastically stable measures.

Stable measures are uniquely determined by boundary conditions.

The study bridges finite and infinite-dimensional stochastic stability theories.

Abstract

In finite-dimensional dynamical systems, stochastic stability provides the selection of physical relevant measures from the myriad invariant measures of conservative systems. That this might also apply to infinite-dimensional systems is the inspiration for this work. As an example the 2D Euler equation is studied. Among other results this study suggests that the coherent structures observed in 2D hydrodynamics are associated to configurations that maximize stochastically stable measures uniquely determined by the boundary conditions in mode space.