Mellon Fellowship colloquium: Problems in Two-Phase Magnetohydrodynamic Equations

We consider the motion of two incompressible, conductive fluids in a magnetic field. The viscosity and surface tension are considered. The work includes the existence of varifold solutions, strong solutions, and their weak-strong uniqueness. To obtain a varifold solution, we use the Galerkin approximation and the weak convergence method. To find a strong solution, we use the Hanzawa transformation and the theory of contraction mappings. The weak-strong uniqueness is obtained by estimating the error between strong and varifold solutions using the relative entropy method.