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 volumepreserving diffeomorphism group $\text{SDiff}(M)$ on smooth exact divergencefree vector fields on a closed threedimensional manifold $M$. More precisely, any regular $C^1$ functional defined on the space of $C^\infty$ (more generally, $C^k$, $k\ge 4$) exact divergencefree vector fields and invariant under arbitrary volumepreserving 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 ArnoldKhesin'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 divergencefree vector fields, the PoincaréBirkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.06008
 arXiv:
 arXiv:2003.06008
 Bibcode:
 2020arXiv200306008K
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Mathematics  Dynamical Systems
 EPrint:
 15 pages