2026/07/23 by Gennaro Ciampa, Renato Lucà
Mathematics · #math.AP
The order of Theorems 1.3 and 1.5 has been reversed. The former Theorem 1.3, now Theorem 1.4, has been strengthened. Minor revisions and typographical corrections have also been made
arxiv created 2026/07/29 · arxiv updated 2026/07/30
We prove that given any sufficiently small and regular solution B of the stationary Euler equations there exists an infinite dimensional family of solutions (u,b) of the non-resistive magnetohydrodynamics equations (MHD) that relax to (0, B). More precisely, (u,b) → (0, B) exponentially fast as t → +∞. This family may be viewed as lying in the stable manifold of the non-resistive MHD equations around the equilibrium state (0, B). The problem whether it actually coincides with the stable manifold remains open. As a byproduct of our result, we provide a large class of global regular solutions of the non-resistive MHD equations. Another consequence is that any sufficiently small and regular solution of the stationary Euler equation is (non-trivially) topologically accessible via MHD from a large class of magnetic fields according to the definition of Moffatt and, in this scenario, the topology of the magnetic lines is (entirely) preserved in the limit t → + ∞.