Characterize two-sided mixing laws in three-dimensional stationary flows

Determine when estimates for the flow Jacobian of three-dimensional stationary flows yield two-sided functional and geometric mixing laws on the relevant regular or chaotic dynamical components.

Background

The paper identifies three-dimensional stationary flows as a major unresolved extension because stationarity does not generally produce an integrable foliation by streamlines. Steady ABC Euler flows may contain chaotic particle trajectories, while other regions may be organized by invariant tori.

The authors emphasize that positive stretching alone is insufficient to establish decay of a prescribed scalar mix-norm: recurrence, folding, and cancellation must also be controlled. The open problem is therefore to identify conditions under which Jacobian estimates imply matching upper and lower bounds for both functional and geometric mixing.