Quasineutral limit of the pressureless Euler-Poisson equation for ions
In this paper, we consider the quasineutral limit of the Euler-Poisson equation for a clod, ion-acoustic plasma when the Debye length tends to zero. When the ion-acoustic plasma is cold, the Euler-Poisson equation is pressureless and hence fails to be Friedrich symmetrisable, which excludes the application of the classical energy estimates method. This brings new difficulties in proving uniform estimates independent of $\varepsilon$. The main novelty in this article is to introduce new $\varepsilon$-weighted norms of the unknowns and to combine energy estimates in different levels with weights depending on $\varepsilon$. Finally, that the quasineutral regimes are the incompressible Euler equations is proven for well prepared initial data.