The Lie-Poisson structure of the LAE-$α$ equation

TL;DR

The paper establishes a precise Hamiltonian (Lie-Poisson) formulation for the averaged Euler equation LAE-$\alpha$ under Dirichlet, Neumann, and mixed boundary conditions, showing the time $t$-map is canonical with respect to a Lie-Poisson bracket arising from a non-smooth reduction on the corresponding diffeomorphism groups. It develops a geometric framework based on $H^1$-like weak Riemannian metrics on diffeomorphism groups, derives a global geodesic spray and connector, and demonstrates that geodesic flow corresponds to LAE-$\alpha$ dynamics for the boundary-condition–dependent groups. The authors define a Lie-Poisson bracket on a class of functionals on the tangent bundle, establish derivative formulas, and prove the flow preserves the Poisson structure, yielding a Poisson formulation of the LAE-$\alpha$ equation with lifted maps $F_t$ and $\tilde{F}_t$ that are Poisson maps. They also extend the framework to free-slip and mixed boundary conditions, introducing a correction term $\mathcal{D}^{\alpha}$ to recover key identities and showing the Poisson and Hamiltonian structures persist in these settings.

Abstract

This paper shows that the time $t$ map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding diffeomorphism groups. It is also shown that the geodesic spray for Neumann and mixed boundary conditions is smooth, a result already known for Dirichlet boundary conditions.

Theorems & Definitions (30)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 3.1
• Lemma 3.2
• Proposition 4.1
• Definition 4.2