  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.
