A system of convection-diffusion equations with small diffusion coefficient arising in semiconductor physics (Q921258)

A system of two convection-diffusion equations in one space dimension with Dirichlet boundary conditions arising in semiconductor theory is studied in this paper. The fact that one of the diffusion coefficients \(\epsilon\) is very small raises some interesting features, in particular the existence of internal spatial layers with fast variation and temporal evolution on two time scales. The nonlocal convection terms depend in a nonlinear way on the unknowns, and this is also very important. The limit case \(\epsilon =0\) is considered and it is proved that solutions of the problem tend (in some topology) to the unique solution of the limit problem (subject to inflow boundary conditions). A priori estimates which are independent on \(\epsilon\) are obtained in Section 2. A compactness argument is used in Section 3 to get solutions of the limit problem, and its uniqueness is shown in Section 4.