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

Dengdi Chen, Yan Zheng

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.

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

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.

Paper Structure

This paper contains 22 sections, 335 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Mathematical setting and notations
  4. Outline of the proof and contributions
  5. Fundamental mathematical framework and main results
  6. The RDS framework and the Multiplicative Ergodic Theorem
  7. Basic setup for random dynamics system
  8. Linear cocycles over RDS and the Multiplicative Ergodic Theorem
  9. Positivity of the top Lyapunov exponent
  10. Furstenberg’s criterion with families of unconditionally continuous measures
  11. Contradicting Furstenberg’s criterion
  12. Priori moment estimates and ergodicity
  13. Priori moment estimates and super-Lyapunov function for stochastic Boussinesq equations
  14. Ergodicity of the Markov processes
  15. Malliavin calculus and probabilistic spectral bound
  16. ...and 7 more sections

Theorems & Definitions (32)

  • proof
  • proof : Proof of Theorem \ref{['L1']}
  • proof : Proof of Remark \ref{['1.2']}
  • proof
  • proof
  • proof : Proof of Proposition \ref{['error']}
  • proof
  • proof
  • proof
  • proof
  • ...and 22 more