We present a very simple proof of the global existence of a C-infinity Lang
rangian flow map for the 2D Eular and second-grade fluid equations (on a co
mpact Riemannian manifold with boundary) which has C-infinity dependence on
initial data u(0) in the class of H-s divergence-free vector fields for s
> 2. (C) Elsevier Science Ltd. All rights reserved.