Hydrodynamical models for semiconductors

We consider a class of hydrodynamic models, which arise in the accurate mathematical modeling of semiconductors, to describe high field phenomena or submicronic devices. From the analytical and the numerical point of view, these models form coupled elliptic-hyperbolic systems of balance laws. Therefore discontinuities (shock waves) can occur in the solutions. We show that, under a suitable relaxation process, these models are asymptotically approximated by a new class of systems of drift-diffusion equations, which take in account the energy transport mechanisms.