Dengdi Chen | Searcharxiv

arXiv2026

Lagrangian chaos for the 2D Boussinesq equations with a degenerate random forcing

We demonstrate that Lagrangian flow for the 2D Boussinesq equations under degenerate noise exhibit chaotic behavior characterized by the strict positivity of the top Lyapunov exponent, where the degenerate noise acts only on a few Fourier modes of the temperature equation. To achieve this, we overcome difficulties arising from the degeneracy of noise and its intricate interaction with the nonlinear terms. This is accomplished by introducing a solution-dependent manifold spanning condition to establish probabilistic spectral bound on a cone for the Malliavin matrix associated with the extended system. Additionally, the approximate controllability of the extended system is realized by constructing smooth controls based on shear and cellular flows.

math.DS↗

arXiv2026

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

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.

math.DS↗