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

2012/05/23 by David Lannes, Lannes, David, Felipe Linares +3 · 5 citations
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #Computational Fluid Dynamics and Aerodynamics

paper · pdf · doi:10.48550/arxiv.1205.5080

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.

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• Hydrodynamic limit from kinetic models with massless electrons to the ionic Euler--Poisson system
• Global well-posedness and scattering for the 2D modified Zakharov-Kuznetsov equation
• Existence of wave operators for Zakharov-Kuznetsov equation in two space dimensions

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