  It is well-known that due to the lack of a technique to obtain the a-priori
$L^{\infty}$ estimate of the artificial viscosity solutions of the Cauchy
problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for
semiconductors, where the energy equation is replaced by a pressure-density
relation, over the past three decades, all solutions of this model were
obtained by using the Lax-Friedrichs, Godounov schemes and Glimm scheme for
both the initial-boundary value problem \cite{Zh1,Li} and the Cauchy problem
\cite{MN1,PRV,HLY}; or by using the vanishing artificial viscosity method for
the initial-boundary value problem \cite{Jo,HLYY}. In this paper, the existence
of global entropy solutions, for the Cauchy problem of this model, is proved by
using the vanishing artificial viscosity method. As a by-product, the known
compactness framework \cite{MN2,JR} is applied to show the relaxation limit, as
the relation time $ \tau$ and $\E,\delta$ go to zero, for general pressure
$P(\rho)$.
