On Euler's equation and 'EPDiff'

Abstract

We study a family of approximations to Euler's equation depending on two parameters ε, η ≥ 0. When ε = η = 0 we have Euler's equation and when both are positive we have instances of the class of integro-differential equations called EPDiff in imaging science. These are all geodesic equations on either the full diffeomorphism group Diff H ∞(ℝn) or, if ε = 0, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by (Equation presented) where Lε,η = (I-η2/pΔ)p o (I-1/ε2Delta; o div). All geodesic equations are locally well-posed, and the Lε,η-equation admits solutions for all time if η > 0 and p ≥ (n + 3)/2. We tie together solutions of all these equations by estimates which, however, are only local in time. This approach leads to a new notion of momentum which is transported by the flow and serves as a generalization of vorticity. We also discuss how delta distribution momenta lead to " vortexsolitons", also called "landmarks" in imaging science, and to new numeric approximations to fluids. © American Institute of Mathematical Sciences.