The global-in-time existence and uniqueness of bounded weak solutions to aspinorial matrix drift-diffusion model for semiconductors is proved. Developingthe electron density matrix in the Pauli basis, the coefficients (chargedensity and spin-vector density) satisfy a parabolic $4\times 4$cross-diffusion system. The key idea of the existence proof is to work withdifferent variables: the spin-up and spin-down densities as well as theparallel and perpendicular components of the spin-vector density with respectto the magnetization. In these variables, the diffusion matrix becomesdiagonal. The proofs of the $L^\infty$ estimates are based on Stampacchiatruncation as well as Moser- and Alikakos-type iteration arguments. Themonotonicity of the entropy (or free energy) is also proved. Numericalexperiments in one space dimension using a finite-volume discretizationindicate that the entropy decays exponentially fast to the equilibrium state.
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