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

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.