Magnetic relaxation for the MHD equations via the stable manifold method

Abstract

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, $(u,b)\to (0,B)$ exponentially fast as $t\to +\infty$. 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 $t\to +\infty$.

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