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

TL;DR

This paper proves that the 2D Boussinesq equations with degenerate random forcing exhibit chaotic Lagrangian flow, demonstrated by positive Lyapunov exponents, through advanced probabilistic and control techniques.

Contribution

It introduces a novel approach combining spectral bounds and controllability to establish chaos in a degenerate stochastic PDE system.

Findings

Positive top Lyapunov exponent for the flow

Chaotic behavior persists despite degenerate noise

New probabilistic spectral bound method

Abstract

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.