The helicity uniqueness conjecture in 3d hydrodynamics
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- by Boris Khesin, Daniel Peralta-Salas and Cheng Yang PDF
- Trans. Amer. Math. Soc. 375 (2022), 909-924 Request permission
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 various tools from the theory of dynamical systems, including normal forms for divergence-free vector fields, the Poincaré-Birkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.References
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Additional Information
- Boris Khesin
- Affiliation: Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
- MR Author ID: 238631
- Email: khesin@math.toronto.edu
- Daniel Peralta-Salas
- Affiliation: Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain
- MR Author ID: 648494
- Email: dperalta@icmat.es
- Cheng Yang
- Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada; and the Fields Institute for Research in Mathematical Sciences, Toronto, ON M5T 3J1, Canada
- ORCID: 0000-0002-9996-9465
- Email: yangc74@math.mcmaster.ca
- Received by editor(s): March 13, 2020
- Received by editor(s) in revised form: May 6, 2021
- Published electronically: December 2, 2021
- Additional Notes: The first author was partially supported by a Simons Fellowship and an NSERC research grant. The second author was supported by the grants MTM PID2019-106715GB-C21 (MICINN) and Europa Excelencia EUR2019-103821 (MCIU), and partially supported by the ICMAT–Severo Ochoa grant CEX2019-000904-S. A part of this work was done while the third author was visiting the Instituto de Ciencias Matemáticas (ICMAT) in Spain. The third author is grateful to the ICMAT for its support and kind hospitality
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 909-924
- MSC (2020): Primary 37K65, 70H33, 37N10
- DOI: https://doi.org/10.1090/tran/8483
- MathSciNet review: 4369239