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

Abstract

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 $s>2$.