  We 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 PDEs