You could look at the paper: Groups of Diffeomorphisms and the motion of an incompressible fluid, by Ebin and Marsden.
About two centuries after Euler, in 1966 Arnold gave a geometric reformulation of the classical equations for an imcompressible fluid in terms of the geodesic spray of left invariant metric on an infinite dimensional Lie Group.
Ebin and Marsden promptly employed this reformulation to obtain existence and uniqueness results for these equations on compact oriented riemannian manifolds.
This circle of ideas is one of the other papers product first important application of their collaborationinfinite dimensional manifolds as remarked by Stephen Smale.
By the way, should not the equation contain the time derivative of the unknown $u$?