We consider the EPDiff equation on \({\mathbb {R}}^n\) with the integer-order homogeneous Sobolev inertia operator \(A=(-\Delta )^k\) . We prove that for arbitrary radial initial data and a sign condition on the initial momentum, the corresponding radial velocity solution has \(C^1\) norm that blows up in finite time whenever \(0\le k<n/2+1.\) Our approach is to use Lagrangian coordinates to formulate EPDiff as an ODE on a Banach space, enabling us to use a comparison estimate with the Liouville equation. Along the way we derive the Green function in terms of hypergeometric functions and discuss their properties. This is a step toward proving the general conjecture that the EPDiff equation is globally well-posed for any Sobolev inertia operator of any real order k if and only if \(k\ge n/2+1\) .
