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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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