Relaxation of the isothermal Euler-Poisson system to the drift-diffusion equations

We consider the one-dimensional Euler-Poisson system in the isothermal case , with a friction coefficient epsilon(-1). When epsilon --> 0(+), we show t hat the sequence of entropy-admissible weak solutions constructed in [PRV] converges to the solution to the drift-diffusion equations. We use the scal ing introduced in [MN2], who proved a quite similar result in the isentropi c cast?, using the theory of compensated compactness. On the one hand, this theory cannot be used in our case; on the other hand, exploiting the linea r pressure law, we can give here a much simpler proof by only using the ent ropy inequality and de la Vallee-Poussin criterion of weak compactness in L -1.