In this thesis we study the Riemannian geometry of diffeomorphism groups equipped with a variety of Sobolev-type metrics. Most notably, we consider the group of volume-preserving diffeomorphisms equipped with a weak L^2 metric, whose geodesics correspond to Lagrangian solutions to the Euler equations.
A well known result of Ebin and Marsden states that Lagrangian solutions to the Euler equations, when framed as an initial value problem, are exactly as smooth as their initial conditions. We investigate a similar regularity property for Lagrangian solutions to the Euler equations when framed as a two-point boundary value problem. In particular, we prove that these L^2 geodesics are exactly as smooth as their boundary conditions.
We achieve like results in an array of other settings including 3D axisymmetric ideal fluids, symplectic Euler equations, Euler-alpha equations, and one dimensional integrable systems including the mu-CH and Hunter-Saxton equations.