A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric

In this note, we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric. We showed that for non-Beltrami fields on a three-dimensional compact manifold, there does not exist Eulerian stable flow which is Lagrangian exponential unstable. We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable.