Lagrangian chaos for the 2D Navier-Stokes equations driven by mildly degenerate noise

Abstract

We consider the 2D incompressible Navier-Stokes equations driven by mildly degenerate noise that acts only on finitely many low Fourier modes, a setting that models large-scale stirring. For this system, we prove that the top Lyapunov exponent of the associated Lagrangian flow is strictly positive, thereby establishing Lagrangian chaos. This result is obtained within the framework of random dynamical systems, combining the multiplicative ergodic theorem with the refined Furstenberg criterion of [25]. Unlike the method in [25] for handling highly degenerate noise, this paper develops a unified analytical framework that combines low-mode control, finite-dimensional Malliavin calculus, and dissipation in the high modes. By constructing a finite-dimensional partial Malliavin matrix and proving its non-degeneracy, we avoid the technical complexity of performing Malliavin analysis on the full phase space and simultaneously overcome the degeneracy introduced by the manifold variables. Furthermore, the mildly degenerate forcing gives controllability in the low-frequency subsystem. In the manifold directions, only first-order Lie brackets are needed, which substantially simplifies the Lie-brackets computations.