Shooting problem for the ideal incompressible fluid

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Consider the motion of ideal incompressible fluid in a bounded 2-dimensional domain (or compact surface) $M$. The fluid configuration is defined by positions of all fluid particles, i.e. by an area preserving diffeomorphism $f: M\to M$. The configuration space of the fluid is the group $SDiff(M)$ of all such diffeomorphisms. Any flow $f_t\in SDiff(M)$ (in the absence of external forces) is a geodesic on $SDiff(M)$ in the $L^2$ metric. Geodesic is defined by the initial position $f_0$ (which may be assumed to be the identity map $Id$ and the initial velocity $v_0$, which is a divergence-free vector field in $M$. The (geodesic) exponential map $exp$ assigns to every velocity $v_0$ the map $exp(v_0)=f_1$.

Theorem: For any $f\in SDiff(M)$ there exists $v_0$ such that $f=exp(v_0)$.

This theorem looks superficially like the classical Hopf-Rinow theorem, but in fact they have nothing in common. The proof is based on some new ideas of microlocal and global analysis.