Analysis of a diffuse interface model for two-phase magnetohydrodynamic flows

In this paper, we consider a diffuse interface model which describes the interaction between a magnetic field B and two immiscible, conducting, incompressible fluids of (volume) averaged velocity u. The model consists of the Cahn-Hilliard equation for the order parameter phi with a singular potential coupled with the equations of resistive magnetohydrodynamics for u and B. The resulting evolution system is endowed with initial conditions and suitable boundary conditions. Here we show the existence of a global weak solution which is unique in dimension two. Stronger regularity assumptions on the ini-tial data allow us to prove the existence of a unique global (resp. local) strong solution in two (resp. three) dimensions. In the two dimensional case, the (global) strong solution is strictly separated, namely, phi stays uniformly away from pure phases. This enables us to deduce a continuous dependence esti-mate. Finally, in dimension two, we establish the instantaneous regularization properties of global weak solutions for t > 0. In particular, we show that phi instantaneously satisfies the strict separation property. This result allows us to establish the convergence to a single equilibrium as well as the existence of a global attractor and the validity of the backward uniqueness property.