  In this article we initiate the mathematical study of the dynamics of a
system of nonlinear Partial Differential Equations modelling the motion of
incompressible, isothermal and conducting modified bipolar fluids in presence
of magnetic field. We mainly prove the existence of weak solutions to the
model. We also prove the existence of a trajectory attractor to the translation
semigroup acting on the trajectories of the set of weak solutions and that of
external forces. Some results concerning the structure of this trajectory
attractor are also given. The results from this paper may be useful in the
investigation of some system of PDEs arising from the coupling of
incompressible fluids of $p$-structure and the Maxwell equations.
