Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations Original Research Article

We present a very simple proof of the global existence of a C∞ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has C∞ dependence on initial data u0 in the class of Hs divergence-free vector fields for s > 2.