Existence of smooth mixing flows on the torus

Establish whether there exist smooth divergence-free flows on the torus that are mixing in the ergodic-theoretic sense, either time-independent or time-periodic, thereby providing deterministic smooth examples with quantitative mixing properties on periodic domains.

Background

Quantitative mixing properties of flows directly control dissipation enhancement rates for advection–diffusion and related equations. While many analyses assume the presence of mixing with a specified rate, constructing explicit smooth deterministic flows on the torus that are mixing (particularly with strong rates) has proved challenging.

There has been progress via deterministic constructions with weaker regularity or special structures and via random models that are more flexible, but the existence of smooth, time-independent or time-periodic mixing flows on the torus remains unresolved.

References

Unfortunately, the question of whether smooth, time-independent or time-periodic, mixing flows exist on the torus is still an open problem, though advances have been made in recent years (see for example~\cites{YaoZlatos17, AlbertiCrippaEA19,EZ19,MHSW22,ELM23} for deterministic constructions and ~\cites{BBPS22,BCZG22} in the more flexible and powerful random setting).

— Enhanced dissipation by advection and applications to PDEs (2501.17695 - Mazzucato et al., 29 Jan 2025) in Subsubsection 'Enhanced dissipation by mixing vector fields' in Section 'Enhanced dissipation through quantitative analysis'