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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 ∞ (R n ) 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 . Alldoi:10.3934/jgm.2013.5.319 fatcat:frliqsmjpvcvrhe6a7fcggk4ya