Year: 2025
Type: preprint
Abstract: Helicity is a fundamental conserved quantity in physical systems governed by vector fields whose evolution is described by volume-preserving transformations on a three-manifold. Notable examples include inviscid, incompressible fluid flows, modeled by the three-dimensional Euler equations, and conducting plasmas, described by the magnetohydrodynamics (MHD) equations. A key property of helicity is its invariance under volume-preserving diffeomorphisms. In an influential article from 1973, Arnold, having provided an ergodic interpretation of helicity as the "asymptotic Hopf invariant", posed the question of whether this invariance persists under volume-preserving homeomorphisms. More generally, he asked whether helicity can be extended to topological volume-preserving flows. We answer both questions affirmatively for flows without rest points. Our approach reformulates Arnold's question in the framework of what we call $C^0$ Hamiltonian structures. This perspective enables us to leverage recent developments in $C^0$ symplectic geometry, particularly results concerning the algebraic structure of the group of area-preserving homeomorphisms.
Source: arXiv (Cornell University)
Authors: Edtmair, Oliver, Seyfaddini, Sobhan
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6929424859
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