A diffuse interface model for incompressible two-phase magnetohydrodynamic flows
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We introduce a diffuse interface model which describes the interaction
between a magnetic field and two partly miscible, conducting,
incompressible fluids. The model consists of the Cahn-Hilliard
equation for the relative difference of the fluid concentrations
coupled with the equations of resistive magnetohydrodynamics for the
volume averaged velocity and for the magnetic field. The resulting
evolution system is endowed with suitable initial and boundary
conditions. We will mostly focus on the longtime behavior of
solutions. More precisely, in dimension two, we will show that we can
define a dissipative dynamical system on a finite energy phase space
which has the global attractor. Moreover, the backward uniqueness
property holds on the global attractor and any weak solution does
converge to a single equilibrium. We shall also discuss some issues in
dimension three and propose some open problems.