Contact type solutions and non-mixing of the 3D Euler equations

TL;DR

This paper demonstrates that the 3D Euler equations do not exhibit mixing behavior on certain manifolds when considering smooth volume-preserving flows with fixed helicity and high energy, using contact and symplectic invariants.

Contribution

It introduces a new framework linking contact/symplectic geometry invariants to Euler solutions, expanding the application of contact topology in hydrodynamics beyond stationary cases.

Findings

Euler equations are non-mixing on closed 3-manifolds under specified conditions.

Spectral invariants from Floer theory serve as new integrals of motion.

Countably many invariant open sets are constructed in the phase space.

Abstract

We prove that on any closed Riemannian three-manifold $(M,g)$ the time-dependent Euler equations are non-mixing on the space of smooth volume-preserving vector fields endowed with the $C^1$-topology, for any fixed helicity and large enough energy, solving a problem posed by Khesin, Misiolek, and Shnirelman. To prove this, we introduce a new framework that assigns contact/symplectic geometry invariants to large sets of time-dependent solutions to the Euler equations on any 3-manifold with an arbitrary fixed metric. This greatly broadens the scope of contact topological methods in hydrodynamics, which so far have had applications only for stationary solutions and without fixing the ambient metric. We further use this framework to prove that spectral invariants obtained from Floer theory, concretely embedded contact homology, define new non-trivial continuous first integrals of the Euler equations in certain regions of the phase space endowed with the $C^{1,s}$-topology, producing countably many disjoint invariant open sets.