The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation

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.

Locations

• Progress in nonlinear differential equations and their applications - View
• arXiv (Cornell University) - View - PDF