2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143234This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontrivial because in Eulerian representation, the time t maps need not be C^1 from the Sobolev class H^s to itself (where s > (n/2) + 1). The idea of how this difficulty is overcome is to exploit the fact that one does have smoothness in the Lagrangian representation and then carefully perform a Lie-Poisson reduction procedure.24 pagesAnalysis of PDEsDifferential Geometry35Q35; 53D17; 53D22; 53D25; 58B20; 58B25; 58D05;76B99text