We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posedness of the system in both space dimensions two and three. Regardless of the size of the initial data, we first prove the global well-posedness of the Cauchy problem when the space dimension is two. However, in space dimension three, we construct global weak-solutions à la Leray, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. Our results extend Giga-Yoshida (1984)  ones to the space dimension two, and improve them in terms of requiring less regularity on the velocity fields.
"Global well posedness for a two-fluid model." Differential Integral Equations 31 (3/4) 187 - 214, March/April 2018.