On the chaotic behavior of the Lagrangian flow of the 2D Navier-Stokes system with bounded degenerate noise

TL;DR

This paper demonstrates that the Lagrangian flow of a 2D Navier-Stokes system with bounded, degenerate noise exhibits chaos, evidenced by a positive top Lyapunov exponent, through a novel abstract controllability-based approach.

Contribution

It introduces a new method linking controllability of deterministic systems to chaos in stochastic fluid dynamics models.

Findings

Lagrangian flow shows chaotic behavior with positive Lyapunov exponent.

Chaos persists under bounded, degenerate forcing on Fourier modes.

New abstract result connects controllability to Lyapunov exponent positivity.

Abstract

We consider a fluid governed by the randomly forced 2D Navier-Stokes system. It is assumed that the force is bounded, acts directly only on a small number of Fourier modes, and satisfies some natural decomposability and observability properties. Under these assumptions, we show that the Lagrangian flow associated with the random fluid exhibits chaotic behavior characterized by the strict positivity of the top Lyapunov exponent. To achieve this, we introduce a new abstract result that allows to derive positivity of the top Lyapunov exponent from controllability properties of the underlying deterministic system.