Fluid mechanical approximation to the degenerated drift-diffusion system from the compressible Navier-Stokes-Poisson system

We consider the zero relaxation time approximation to the compressible Navier-Stokes-Poisson system to derive the mono-polar drift-diffusion system of degenerated type. To construct the approximation solution, we introduce a regularized, damped, viscous, compressible Navier-Stokes-Poisson equation; as the relaxation time τ → 0, the weak solution converges to that of the degenerated elliptic parabolic system. The entropy functional is rigorously derived from the energy inequality of the approximation system. We employ the argument developed in the theory of weak solutions in the compressible Navier-Stokes-Poisson system with the regularized pressure.