Analysis of a coupled spin drift–diffusion Maxwell–Landau–Lifshitz system

The existence of global weak solutions to a coupled spin drift–diffusion and Maxwell–Landau–Lifshitz system is proved. The equations are considered in a two-dimensional magnetic layer structure and are supplemented with Dirichlet–Neumann boundary conditions. The spin drift–diffusion model for the charge density and spin density vector is the diffusion limit of a spinorial Boltzmann equation for a vanishing spin polarization constant. The Maxwell–Landau–Lifshitz system consists of the time-dependent Maxwell equations for the electric and magnetic fields and of the Landau–Lifshitz–Gilbert equation for the local magnetization, involving the interaction between magnetization and spin density vector. The existence proof is based on a regularization procedure, L2L2-type estimates, and Moser-type iterations which yield the boundedness of the charge and spin densities. Furthermore, the free energy is shown to be nonincreasing in time if the magnetization–spin interaction constant in the Landau–Lifshitz equation is sufficiently small.