Arnold discovered that ideal incompressible fluids traced out geodesics in the group of volume-preserving diffeomorphisms. Until recently, it was hoped that negative curvature in this group would lead to exponential Lagrangian instability of fluid flows. I will briefly mention the counterexamples which show that this does not work.

I will mainly discuss a new result which relates Lagrangian stability of a steady flow to Eulerian stability: a steady incompressible flow has exponentially growing Jacobi fields along the corresponding geodesic if and only if the linearized Euler equation has exponentially growing solutions. This is true in all Sobolev norms, and I will present a relation between the long-time asymptotics of Sobolev norms of Lagrangian perturbations and Eulerian perturbations, in both two and three dimensions.