Geometric, Variational, and Bracket Descriptions of Fluid Motion with Open Boundaries

The paper’s Proposition 3.6 asserts the extended Lie–Poisson bracket identity for open fluids and claims equivalence with the PDE system and boundary conditions (3.12)–(3.13); it also gives the precise bulk+boundary split (3.24) used to expose the boundary ports J, j_ρ, j_s . The candidate solution derives the same identity directly from the PDEs via a transport/divergence identity and integration by parts, and then proves the converse by testing with linear functionals. This matches the paper’s result but uses a different (direct, PDE-level) proof style; the paper’s Eulerian proof is sketched via geometric reduction and “taking time derivatives on solutions,” while the material-side equivalence is stated and justified explicitly (3.18)–(3.22) . Minor sign-convention remarks in the candidate are consistent with the paper’s conventions. Overall, both are correct; the model fills in some details (the converse direction in Eulerian form) that the paper states but does not detail.