  A system of drift-diffusion equations with electric field under Dirichlet
boundary conditions is analyzed. The system of strongly coupled parabolic
equations for particle density and spin density vector describes the
spin-polarized semi-classical electron transport in ferromagnetic
semiconductors. The presence of a nonconstant and nonsmooth magnetization
vector, solution of the Landau-Lifshitz equation, causes the diffusion matrix
to be dependent from space and time and to have in general poor regularity
properties, thus making the analysis challenging. To partially overcome the
analytical difficulties the velocity saturation hypothesis is made, which
results in a bounded drift velocity. The global-in-time existence and
uniqueness of weak solutions is shown by means of a semi-discretization in
time, which yields an elliptic semilinear problem, and a quadratic entropy
inequality, which allow for the limit of vanishing time step size. The
convergence of the weak solutions to the steady state, under some restrictions
on the parameters and data, is shown. Finally the higher regularity of
solutions for a smooth magnetization in two space dimensions is shown through a
diagonalization argument, which allows to get rid of the cross diffusion terms
in the fluid equations, and the iterative application of Gagliardo-Nirenberg
inequalities and a generalized version of Aubin lemma.
