Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

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Abstract

This paper is concerned with the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (ofn or of an n-dimensional manifold M). It focuses on Lagrangians that are quadratic in the velocity fields and their first derivatives, that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H 1 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 for shallow water waves in one dimension. The corresponding equations for the volume-preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible fluids.

The first author is grateful for support by US DOE under contract W-7405-ENG-36 for Los Alamos National Laboratory, and Office of Science ASCAR/AMS/MICS. The research of the second author was partially supported by the California Institute of Technology, by the National Science Foundation through NSF grant DMS-0204474, and by Air Force contract F49620-02-1-0176.
To Alan Weinstein on the occasion of his 60th birthday.