We consider a system of two convection-diffusion equations with a small diffusion parameter in one space dimension subject to Dirichlet boundary conditions. The system governs the evolution of the flow of electrons and holes in semiconductor devices on the dielectrical relaxation time scale. The equations are coupled by nonlinear, nonlocal (electric field driven) convection terms. We prove the convergence (in suitable topologies) of the solutions of the diffusion-convection problem to the unique solution of the convective limit problem (subject to inflow boundary conditions) as the diffusion coefficient tends to zero.
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