Quasineutral limit of the pressureless Euler-Poisson equation for ions

Abstract

Abstract. 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 ionacoustic 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 ε. The main novelty in this article is to introduce new ε-weighted norms of the unknowns and to combine energy estimates in different levels with weights depending on ε. Finally, that the quasineutral regimes are the incompressible Euler equations is proven for well prepared initial data.

The initial layer problem of the Euler–Poisson in collisionless plasma

“…Ju et al [4] proved the quasi‐neutral limit and vanishing viscosity limit of the isothermal Navier–Stokes–Poisson system with boundary. The other related results on the Euler–Poisson equations can be found in earlier studies [5–15].…”

Long Wavelength Limit for the Quantum Euler--Poisson Equation

“…Han-Kwan [12] also introduced a long wave scaling for the Vlasov-Poisson equation and derived the KdV equation in 1D and the Zakharov-Kuznetsov equation in 3D using the modulated energy method. For other studies for the Euler-Poisson system or related models, the interested readers may refer to [2,3,5,23,24,28], to list only a few. For derivation of the KdV equation from the water waves without surface tension, see [29] and the references therein.…”

Quasineutral limit for the quantum Navier-Stokes-Poisson equations

“…To the author's best knowledge, the first quasi-neutral limit is on the Euler-Poisson equation for ions with positive ion temperature by Cordier and Grenier [5]. This result was recently generalized to the Euler-Poisson equation with zero ion temperature for cold plasmas in [31]. Since then, quasi-neutral limit results have been obtained for various hydrodynamic models in plasmas.…”