2025/03/18 by Agresti, Antonio
#60H15 #76M35 #76U60 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 37H15 #Probability (math.PR) #Secondary: 35Q86
paper · doi:10.48550/arxiv.2503.14658
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.