On Euler's equation and `EPDiff'

Abstract: We study a family of approximations to Euler's equation depending on two parameters $\varepsilon,\eta \ge 0$. When $\varepsilon=\eta=0$ 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 $\operatorname{Diff}{H^\infty}(\mathbb R^n)$ or, if $\varepsilon = 0$, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by $$ |v|{\varepsilon,\eta} = \int_{\mathbb R^n} <L_{\varepsilon,\eta} v, v> dx $$ where $L_{\varepsilon,\eta} = (I-\tfrac{\eta^2}{p} \triangle)^p \circ (I-\tfrac1{\varepsilon^2} \nabla \circ \div)$. All geodesic equations are locally well-posed, and the $L_{\varepsilon,\eta}$-equation admits solutions for all time if $\eta>0$ and $p\ge (n+3)/2$. 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.