Critical-index global existence conjecture for EPDiff equations

Let $n$ be the spatial dimension, let $k\in\mathbb{N}$, and consider the Euler–Arnold equation with metric parameter $k$ on $\operatorname{Diff}(\mathbb{R}^n)$. The corresponding metric is denoted by $G^k$.

Critical-index conjecture. All solutions exist for all time if $k\geq \frac{n}{2}+1$, whereas for $k<\frac{n}{2}+1$ there are always some solutions that break down in finite time. Consequently, $(\operatorname{Diff}(\mathbb{R}^n),G^k)$ is geodesically complete if and only if $k\geq \frac{n}{2}+1$.

This conjecture identifies the Sobolev critical index separating global well-posedness from finite-time breakdown. The source notes that the two-dimensional critical case $k=2$ has been settled for radial solutions by different methods, while finite-time breakdown for higher integer values of $k<\frac{n}{2}+1$ remains to be proved in dimensions $n>3$.