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 $ {\boldsymbol {B}} $ and two immiscible, conducting, incompressible fluids of (volume) averaged velocity $ {\boldsymbol {u}} $. The model consists of the Cahn–Hilliard equation for the order parameter $ \varphi $ with a singular potential coupled with the equations of resistive magnetohydrodynamics for $ {\boldsymbol {u}} $ and $ {\boldsymbol {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 initial 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, $ \varphi $ stays uniformly away from pure phases. This enables us to deduce a continuous dependence estimate. Finally, in dimension two, we establish the instantaneous regularization properties of global weak solutions for $ t>0 $. In particular, we show that $ \varphi $ 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. [ABSTRACT FROM AUTHOR]