The helicity uniqueness conjecture in 3d hydrodynamics

Abstract

We prove that the helicity is the only regular Casimir function for the coadjoint action of the volume-preserving diffeomorphism group $\text{SDiff}(M)$ on smooth exact divergence-free vector fields on a closed three-dimensional manifold $M$. More precisely, any regular $C^1$ functional defined on the space of $C^\infty$ (more generally, $C^k$, $k\ge 4$) exact divergence-free vector fields and invariant under arbitrary volume-preserving diffeomorphisms can be expressed as a $C^1$ function of the helicity. This gives a complete description of Casimirs for adjoint and coadjoint actions of $\text{SDiff}(M)$ in 3D and completes the proof of Arnold-Khesin's 1998 conjecture for a manifold $M$ with trivial first homology group. Our proofs make use of different tools from the theory of dynamical systems, including normal forms for divergence-free vector fields, the Poincar\'e-Birkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.