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Lagrangian chaos and unique ergodicity for stochastic primitive equations

Antonio Agresti

TL;DR

This work proves Lagrangian chaos for the three-dimensional stochastic primitive equations by establishing a strictly positive top Lyapunov exponent, $\lambda_+>0$, for the random Lagrangian flow and its projective dynamics. The authors develop new high-order energy estimates in anisotropic Sobolev spaces and demonstrate integrability and regularity of invariant measures, which, combined with a strong Feller property and approximate controllability, yield uniqueness of invariant measures and ergodicity. The approach extends the BBPS framework to a physically realistic, energy-supercritical 3D fluid model with nondegenerate noise, and it shows gradient growth for passive scalars advected by the flow. The results have significant implications for mixing, long-time statistical behavior, and the ergodic description of stochastic geophysical flows.

Abstract

We show that the Lagrangian flow associated with the stochastic 3D primitive equations (PEs) with non-degenerate noise is chaotic, i.e., the corresponding top Lyapunov exponent is strictly positive almost surely. This result builds on the landmark work by Bedrossian, Blumenthal, and Punshon-Smith on Lagrangian chaos in stochastic fluid mechanics. Our primary contribution is establishing an instance where Lagrangian chaos can be proven for a fluid flow with supercritical energy, a key characteristic of 3D fluid dynamics. For the 3D PEs, establishing the existence of the top Lyapunov exponent is already a challenging task. We address this difficulty by deriving new estimates for the invariant measures of the 3D PEs, which capture the anisotropic smoothing in the dynamics of the PEs. As a by-product of our results, we also obtain the first uniqueness result for invariant measures of stochastic PEs.

Lagrangian chaos and unique ergodicity for stochastic primitive equations

TL;DR

This work proves Lagrangian chaos for the three-dimensional stochastic primitive equations by establishing a strictly positive top Lyapunov exponent, , for the random Lagrangian flow and its projective dynamics. The authors develop new high-order energy estimates in anisotropic Sobolev spaces and demonstrate integrability and regularity of invariant measures, which, combined with a strong Feller property and approximate controllability, yield uniqueness of invariant measures and ergodicity. The approach extends the BBPS framework to a physically realistic, energy-supercritical 3D fluid model with nondegenerate noise, and it shows gradient growth for passive scalars advected by the flow. The results have significant implications for mixing, long-time statistical behavior, and the ergodic description of stochastic geophysical flows.

Abstract

We show that the Lagrangian flow associated with the stochastic 3D primitive equations (PEs) with non-degenerate noise is chaotic, i.e., the corresponding top Lyapunov exponent is strictly positive almost surely. This result builds on the landmark work by Bedrossian, Blumenthal, and Punshon-Smith on Lagrangian chaos in stochastic fluid mechanics. Our primary contribution is establishing an instance where Lagrangian chaos can be proven for a fluid flow with supercritical energy, a key characteristic of 3D fluid dynamics. For the 3D PEs, establishing the existence of the top Lyapunov exponent is already a challenging task. We address this difficulty by deriving new estimates for the invariant measures of the 3D PEs, which capture the anisotropic smoothing in the dynamics of the PEs. As a by-product of our results, we also obtain the first uniqueness result for invariant measures of stochastic PEs.

Paper Structure

This paper contains 38 sections, 47 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. Background and contribution
  3. Future challenges
  4. Overview
  5. Preliminaries and set-up
  6. Reformulation and function analytic setting
  7. Probabilistic set-up and $H^\sigma$ solutions
  8. $H^\sigma$-solutions
  9. Probabilistic set-up
  10. Statement of the main results and strategy
  11. Global well-posedness in $H^\sigma$ and unique ergodicity
  12. Lagrangian chaos for stochastic PEs
  13. Outline and proofs of Theorems \ref{['t:positive_lyapunov_exponent']} and \ref{['t:positive_lyapunov_exponent_II']}
  14. Integrability of invariant measures
  15. Auxiliary results
  16. ...and 23 more sections

Key Result

Theorem 1.1

Under suitable smoothness and nondegeneracy assumptions on the noise $QW_t$, there exists a deterministic constant $\lambda_+>0$ such that for all sufficiently regular initial data $v$ the (random) Lagrangian flow $\phi^t$ associated with the stochastic PEs eq:primitive_full is chaotic, i.e., satisfies

Theorems & Definitions (82)

  • Theorem 1.1: Lagrangian chaos PEs -- Informal statement
  • Theorem 1.2: Uniqueness and regularity of invariant measures -- Informal statement
  • Definition 2.1: Local, unique and global $H^\sigma$-solutions
  • Proposition 3.1: Global well-posedness in higher-order Sobolev spaces
  • Proposition 3.2: Regularity of invariant measures
  • Theorem 3.3: Uniqueness and support of invariant measures
  • Example 3.4
  • Theorem 3.5: Positive top Lyapunov exponent
  • Theorem 3.6: Positive top Lyapunov exponent II
  • Corollary 3.7: Gradient growth for passive scalars
  • ...and 72 more