On Euler's equation and `EPDiff'

Original · EN

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 Diffₕ∞(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 vε,η = ∫ᵣₙ <Lε,η v, v> dx where Lε,η = (I-η²p)ᵖ ∘ (I-1ε² ∇ ∘ ÷). 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 "vortex-solitons", also called "landmarks" in imaging science, and to new numeric approximations to fluids.