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On a problem of magnetohydrodynamics in a multi-connected domain. (English) Zbl 1402.35227
Summary: We consider the following problem in the MHD approximation: the vessel \(\Omega _{1} \subset \Omega \) is filled with an incompressible, electrically conducting fluid, and is surrounded by a dielectric or by vacuum, occupying the bounded domain \(\Omega _{2}=\Omega \setminus \Omega _{1}\). In \(\Omega \) we have a magnetic and electric field and the external surface \(S=\partial \Omega \) is an ideal conductor. The emphasis in the paper is on when \(\Omega \) 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.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
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