On a problem of magnetohydrodynamics in a multi-connected domain

We consider the following problem in the MHD approximation: the vessel Ω1⊂ΩΩ1⊂Ω is filled with an incompressible, electrically conducting fluid, and is surrounded by a dielectric or by vacuum, occupying the bounded domain Ω2=Ω∖Ω1Ω2=Ω∖Ω1. In ΩΩ we have a magnetic and electric field and the external surface S=∂ΩS=∂Ω is an ideal conductor. The emphasis in the paper is on when ΩΩ is not simply connected, in which case the MHD system is degenerate. We use Hodge-type decomposition theorems to obtain strong solutions locally in time or global for small enough initial data, and a linearization principle for the stability of a stationary solution.