ASYMPTOTIC ESTIMATES FOR THE MULTIDIMENSIONAL ELECTRO-DIFFUSION EQUATIONS

The drift-diffusion equations can be studied in the framework of singu lar perturbation analysis as a small parameter characterizing the devi ce goes to zero. A formal asymptotic expansion, which includes interna l (and eventually boundary) layer terms can be derived by standard tec hniques of asymptotic analysis. We present in this paper L-2 estimates for the difference between the solutions of the full system and the f irst term of the expansion which are valid for multi-dimensional devic es close to equilibrium. These estimates are based on a uniform monoto ne property of the equations with respect to the small parameter and o n a L-infinity bound for the gradient of solutions of equations under divergence form whose coefficients are bounded, and have derivatives i n one direction which are bounded with respect to all variables.