Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation

We study the dynamics of measure-valued solutions of what we call the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of Rn or an n-dimensional manifold M). Our main focus will be on the case of quadratic Lagrangians; that is, on geodesic motion on the diffeomorphism group with respect to the right invariant Sobolev H metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations in one dimension. The corresponding equations for the volume preserving diffeomorphism group are the well-known LAE (Lagrangian averaged Euler) equations for incompressible fluids. We first show that the EPDiff equations are generated by a smooth vector field on the diffeomorphism group for sufficiently smooth solutions. This is analogous to known results for incompressible fluids—both the Euler equations and the LAE equations—and it shows that for sufficiently smooth solutions, the equations are well-posed for short time. In fact, numerical evidence suggests