### Relaxation of the Isothermal Euler-Poisson System to the Drift-Diffusion Equations

S. Junca and M. Rascle

Abstract: We consider the one dimensional Euler-Poisson system, in the { \em isothermal case}, with a {\em friction coefficient} $\epsi^{-1}.$ When $\epsi \rightarrow 0_+$, we show that the sequence of entropy-admissible weak solutions constructed in \cite{PRV} converges to the solution to the drift-diffusion equations. We use the scaling introduced in \cite{MN2}, who proved a quite similar result in the {\em isentropic} case, using the theory of compensated compactness. On 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 Vall\'ee-Poussin criterion of weak compactness in $L^1.$

Paper:
Available as PostScript.
Author(s):
S. Junca , <junca@math.unice.fr>
M. Rascle , <rascle@math.unice.fr>
Publishing information:
To appear Quart. J. Appl. Math.