2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59529We present a very simple proof of the global existence of a $C^\infty$ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has $C^\infty$ dependence on initial data $u_0$ in the class of $H^s$ divergence-free vector fields for $s>2$.Analysis of PDEs35Q35text