Lagrangian chaos and unique ergodicity for stochastic primitive equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912364304007168 |
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| author | Agresti, Antonio |
| author_facet | Agresti, Antonio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lagrangian chaos and unique ergodicity for stochastic primitive equations Agresti, Antonio Probability Mathematical Physics Analysis of PDEs Primary: 37H15, Secondary: 35Q86, 60H15, 76M35, 76U60 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. |
| title | Lagrangian chaos and unique ergodicity for stochastic primitive equations |
| topic | Probability Mathematical Physics Analysis of PDEs Primary: 37H15, Secondary: 35Q86, 60H15, 76M35, 76U60 |
| url | https://arxiv.org/abs/2503.14658 |