  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-posdness 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
\`a 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.
