The depth-averaged 3D Euler equations for shallow water flow are well approximated by 2D equations of geodesic motion for a certain Sobolev
norm. The initial value problem for these equations produces filamentary soliton-like solutions that are measure-valued, that is, they are supported on curves that evolve in the plane. Numerics shows these filamentary
solutions emerge and dominate the initial value problem for any confined initial velocity distribution. These solutions have three interesting properties: they superpose, they form an invariant manifold and their
nonlinear interactions allow them to reconnect with each other in 2D. This phenomenon of reconnection is also observed for oceanic internal waves by synthetic aperture radar measurements taken from the space shuttle. Thus,
these filamentary solutions of the depth-averaged 3D Euler equations provide a simplified representation for studying evolving 2D arrays of
interacting internal waves.
Remarkably, the same family of geodesic equations also emerges for image processing in the template matching approach to computational
anatomy. Here, a measure-valued solution at two times corresponds to the "cartoons" of an image and its target image under the template-matching
map.
The existence of these measure-valued solutions of geodesic flow is guaranteed -- with any Sobolev norm, and in any number of spatial
dimensions -- because the solution ansatz is a momentum map for the action of diffeomorphisms on the measure-valued support set of the solutions.
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