A Basis of Casimirs in 3D Magnetohydrodynamics. (9th February 2020)

• Record Type: Journal Article
• Authors: Khesin, Boris; Peralta-Salas, Daniel; Yang, Cheng
• Is Part Of: International mathematics research notices. Volume 2021:Number 18(2021)
• Page Start: 13645
• Page End: 13660
• Publication Date: 2020-02-09
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnz393
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We prove that any regular Casimir in 3D magnetohydrodynamics (MHD) is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\textrm{SDiff}(M)\ltimes \mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.