Analysis of a drift–diffusion model with velocity saturation for spin-polarized transport in semiconductors

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 on 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.