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 ${\varepsilon ^{ - 1}}$. When $\varepsilon \to {0_ + }$, we show that the sequence of entropy-admissible weak solutions constructed in [PRV] converges to the solution to the drift-diffusion equations. We use the scaling introduced in [MN2], who proved a quite similar result in the isentropic case, using the theory of compensated compactness. On the one hand, this theory cannot be used in our case; on the other hand, exploiting the linear pressure law, we can give here a much simpler proof by only using the entropy inequality and de la Vallée-Poussin criterion of weak compactness in ${L^{1}}$.