The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation

TL;DR

This paper rigorously derives the Zakharov-Kuznetsov equation from the Euler-Poisson system in magnetized plasmas, establishing local well-posedness and analyzing the long-wave limit in cold and warm plasma cases.

Contribution

It provides a rigorous justification of the Zakharov-Kuznetsov equation from the Euler-Poisson system, including well-posedness and limit analysis in different plasma conditions.

Findings

Proved local well-posedness in 2D and 3D

Derived Zakharov-Kuznetsov as long-wave limit

Extended results to isothermal pressure case

Abstract

We consider in this paper the rigorous justification of the Zakharov-Kuznetsov equation from the Euler-Poisson system for uniformly magnetized plasmas. We first provide a proof of the local well-posedness of the Cauchy problem for the aforementioned system in dimensions two and three. Then we prove that the long-wave small-amplitude limit is described by the Zakharov-Kuznetsov equation. This is done first in the case of cold plasma; we then show how to extend this result in presence of the isothermal pressure term with uniform estimates when this latter goes to zero.