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

Dengdi Chen, Yan Zheng

Abstract

We consider the 2D incompressible Navier-Stokes equations driven by mildly degenerate noise acting only on finitely many low Fourier modes, corresponding to 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 from [26]. Unlike the method in [26] 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 setup of mildly degenerate noise ensures natural controllability in the low-frequency subsystem, while controllability in the manifold directions requires only first order Lie brackets, significantly simplifying the Lie algebraic computations.

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 acting only on finitely many low Fourier modes, corresponding to 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 from [26]. Unlike the method in [26] 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 setup of mildly degenerate noise ensures natural controllability in the low-frequency subsystem, while controllability in the manifold directions requires only first order Lie brackets, significantly simplifying the Lie algebraic computations.

Paper Structure

This paper contains 16 sections, 211 equations.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Setup and notations
  4. Outline of the proof and contributions
  5. Discussion of control construction and error estimates
  6. Discussion of the nondegeneracy of the partial Malliavin matrix
  7. Fundamental mathematical framework and Furstenberg criterion
  8. RDS framework and Multiplicative ergodic theorem
  9. Furstenberg criterion and results
  10. Asymptotic gradient estimate
  11. Malliavin calculus and proof strategy
  12. Splitting of extended system
  13. Construction and cost of control
  14. Error representation and estimates
  15. Nondegeneracy of the partial Malliavin matrix
  16. ...and 1 more sections

Theorems & Definitions (12)

  • proof : Proof of the Remark \ref{['1.2']}
  • proof : Proof of Proposition \ref{['v-estimate-p']}
  • proof
  • proof
  • proof
  • proof : Proof of Proposition \ref{['M-1']}
  • proof : Proof of Lemma \ref{['lower']}
  • proof : Proof of Lemma \ref{['upper']}
  • proof : Proof of Lemma \ref{['spectral bound']}
  • proof
  • ...and 2 more