Under special physical conditions, certain parameters in the quasi-hydrodynamic equations for semiconductors and plasmas are small compared to reference values. Neglecting the corresponding terms, one gets simplified equations which are numerically easier to solve and which contain the important physical informations. The study of the asymptotic limits and their mathematical verification is of great importance to understand the validity of the various models.

In this talk we analyze three asymptotic limits: the relaxation-time limit, the zero-electron-mass limit, and the zero-space-charge limit in the hydrodynamic (Euler-Poisson) equations and the drift-diffusion equations.

The proofs of these results are all based on a priori estimates given by the entropy functional and appropriate compactness methods (compensated compactness, compactness-by-convexity).

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