2023/12/22 by Robert Cardona, Cardona, Robert, Nathan Duignan +3 · 1 citation
Engineering · Mathematics · #35Q31 #53B20 #53C80 (Secondary) #76W05 (Primary) 53C20 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Plasma Physics (physics.plasm-ph) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2312.14368
openalex publication_date 2023/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Ideal magnetohydrodynamic (MHD) equilibria on a Riemannian 3-manifold satisfy the stationary Euler equations for ideal fluids. A stationary solution X admits a large set of ``adapted" metrics in M for which X solves the corresponding MHD equilibrium equations with the same pressure function. We prove different versions of the following statement: an MHD equilibrium with non-constant pressure on a compact three-manifold with or without boundary admits no continuous Killing symmetries for an open and dense set of adapted metrics. This contrasts with the classical conjecture of Grad which loosely states that an MHD equilibrium on a toroidal Euclidean domain in ℝ3 with pressure function foliating the domain with nested toroidal surfaces must admit Euclidean symmetries.