an:05819590
Zbl 1402.35227
Mosconi, Sunra J. N.; Solonnikov, Vsevolod A.
On a problem of magnetohydrodynamics in a multi-connected domain
EN
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 2, 462-478 (2011).
00270022
2011
j
35Q35 35B35 76D03 76W05
magnetohydrodynamics; Navier-Stokes equations; stability; Hodge decomposition; Maxwell equations
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.