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On nonlinear stability of MHD equilibrium figures. (English) Zbl 1284.76412

Fursikov, Andrei V. (ed.) et al., New directions in mathematical fluid mechanics. The Alexander V. Kazhikhov memorial volume. Boston, MA: Birkhäuser (ISBN 978-3-0346-0151-1/hbk). Advances in Mathematical Fluid Mechanics, 301-331 (2010).
Summary: We study the problem of equilibrium figures of electro-conducting fluids. Our first goal is to set correctly the initial boundary value problem for the equations governing both incompressible and compressible flows of electrically conducting fluids with unknown free surface. Notice that when the exterior is a dielectric or a vacuum, attention cannot be confined merely to the region of the conducting fluid; this represents a crucial point for the well-posedness problem. The second goal is to study a new criterion of nonlinear stability of the rest state of a heavy electro-conducting, incompressible or compressible fluid in a section of horizontal layer with rigid plane bottom, and upper unknown free boundary in a vacuum. The new criterion proposes an alternative definition of perturbation, and is deeply related to the unknown motion of the boundary. The third goal is to prove nonlinear stability, in the class of global regular solutions, if the system has non-significant magnetic susceptibility, in absence of surface currents, for large initial data. Kinematic viscosity, magnetic diffusivity, surface tension are only non-negative.
For the entire collection see [Zbl 1182.35002].

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
35M10 PDEs of mixed type
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